Linear algebra is the branch of mathematics concerning linear equations such as:
147-402: Linear maps such as: and their representations in vector spaces and through matrices . Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry , including for defining basic objects such as lines , planes and rotations . Also, functional analysis , a branch of mathematical analysis , may be viewed as
294-406: A 1 e 1 , … , a k e k {\displaystyle a_{1}\mathbf {e} _{1},\ldots ,a_{k}\mathbf {e} _{k}} is a basis of G , for some nonzero integers a 1 , … , a k {\displaystyle a_{1},\ldots ,a_{k}} . For details, see Free abelian group § Subgroups . In
441-506: A i , j v i = ∑ i = 1 n ( ∑ j = 1 n a i , j y j ) v i . {\displaystyle \mathbf {x} =\sum _{j=1}^{n}y_{j}\mathbf {w} _{j}=\sum _{j=1}^{n}y_{j}\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}=\sum _{i=1}^{n}{\biggl (}\sum _{j=1}^{n}a_{i,j}y_{j}{\biggr )}\mathbf {v} _{i}.} The change-of-basis formula results then from
588-416: A k are in F form a linear subspace called the span of S . The span of S is also the intersection of all linear subspaces containing S . In other words, it is the smallest (for the inclusion relation) linear subspace containing S . A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vectors is linearly independent if the only way to express
735-666: A k , b k . But many square-integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis . The geometric notions of an affine space , projective space , convex set , and cone have related notions of basis . An affine basis for an n -dimensional affine space
882-618: A linear extension of f {\displaystyle f} to X , {\displaystyle X,} if it exists, is a linear map F : X → Y {\displaystyle F:X\to Y} defined on X {\displaystyle X} that extends f {\displaystyle f} (meaning that F ( s ) = f ( s ) {\displaystyle F(s)=f(s)} for all s ∈ S {\displaystyle s\in S} ) and takes its values from
1029-427: A = 0 (one constraint), and in that case the solution space is ( x , b ) or equivalently stated, (0, b ) + ( x , 0), (one degree of freedom). The kernel may be expressed as the subspace ( x , 0) < V : the value of x is the freedom in a solution – while the cokernel may be expressed via the map W → R , ( a , b ) ↦ ( a ) {\textstyle (a,b)\mapsto (a)} : given
1176-438: A Hilbert basis (linear programming) . For a probability distribution in R with a probability density function , such as the equidistribution in an n -dimensional ball with respect to Lebesgue measure, it can be shown that n randomly and independently chosen vectors will form a basis with probability one , which is due to the fact that n linearly dependent vectors x 1 , ..., x n in R should satisfy
1323-412: A finite basis is called finite-dimensional . In this case, the finite subset can be taken as B itself to check for linear independence in the above definition. It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing orientation , or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to
1470-469: A linearly independent set L of n elements of V , one may replace n well-chosen elements of S by the elements of L to get a spanning set containing L , having its other elements in S , and having the same number of elements as S . Most properties resulting from the Steinitz exchange lemma remain true when there is no finite spanning set, but their proofs in the infinite case generally require
1617-431: A linearly independent spanning set . Such a linearly independent set that spans a vector space V is called a basis of V . The importance of bases lies in the fact that they are simultaneously minimal generating sets and maximal independent sets. More precisely, if S is a linearly independent set, and T is a spanning set such that S ⊆ T , then there is a basis B such that S ⊆ B ⊆ T . Any two bases of
SECTION 10
#17327808476551764-716: A matrix . This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if A {\displaystyle A} is a real m × n {\displaystyle m\times n} matrix, then f ( x ) = A x {\displaystyle f(\mathbf {x} )=A\mathbf {x} } describes a linear map R n → R m {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}} (see Euclidean space ). Let { v 1 , … , v n } {\displaystyle \{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}} be
1911-604: A multivariate function at a point is the linear map that best approximates the function near that point. The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art . Its use is illustrated in eighteen problems, with two to five equations. Systems of linear equations arose in Europe with
2058-418: A ring ). The multiplicative identity element of this algebra is the identity map id : V → V {\textstyle \operatorname {id} :V\to V} . An endomorphism of V {\textstyle V} that is also an isomorphism is called an automorphism of V {\textstyle V} . The composition of two automorphisms is again an automorphism, and
2205-724: A vector subspace of a real or complex vector space X {\displaystyle X} has a linear extension to all of X . {\displaystyle X.} Indeed, the Hahn–Banach dominated extension theorem even guarantees that when this linear functional f {\displaystyle f} is dominated by some given seminorm p : X → R {\displaystyle p:X\to \mathbb {R} } (meaning that | f ( m ) | ≤ p ( m ) {\displaystyle |f(m)|\leq p(m)} holds for all m {\displaystyle m} in
2352-744: A basis for V {\displaystyle V} . Then every vector v ∈ V {\displaystyle \mathbf {v} \in V} is uniquely determined by the coefficients c 1 , … , c n {\displaystyle c_{1},\ldots ,c_{n}} in the field R {\displaystyle \mathbb {R} } : v = c 1 v 1 + ⋯ + c n v n . {\displaystyle \mathbf {v} =c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n}.} If f : V → W {\textstyle f:V\to W}
2499-488: A basis for W {\displaystyle W} . Then we can represent each vector f ( v j ) {\displaystyle f(\mathbf {v} _{j})} as f ( v j ) = a 1 j w 1 + ⋯ + a m j w m . {\displaystyle f\left(\mathbf {v} _{j}\right)=a_{1j}\mathbf {w} _{1}+\cdots +a_{mj}\mathbf {w} _{m}.} Thus,
2646-448: A basis of R . More generally, if F is a field , the set F n {\displaystyle F^{n}} of n -tuples of elements of F is a vector space for similarly defined addition and scalar multiplication. Let e i = ( 0 , … , 0 , 1 , 0 , … , 0 ) {\displaystyle \mathbf {e} _{i}=(0,\ldots ,0,1,0,\ldots ,0)} be
2793-580: A basis of V . By definition of a basis, every v in V may be written, in a unique way, as v = λ 1 b 1 + ⋯ + λ n b n , {\displaystyle \mathbf {v} =\lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n},} where the coefficients λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are scalars (that is, elements of F ), which are called
2940-446: A difference w – z , and the line segments wz and 0( w − z ) are of the same length and direction. The segments are equipollent . The four-dimensional system H {\displaystyle \mathbb {H} } of quaternions was discovered by W.R. Hamilton in 1843. The term vector was introduced as v = x i + y j + z k representing a point in space. The quaternion difference p – q also produces
3087-508: A finite-dimensional vector space over a field F , and ( v 1 , v 2 , ..., v m ) be a basis of V (thus m is the dimension of V ). By definition of a basis, the map is a bijection from F , the set of the sequences of m elements of F , onto V . This is an isomorphism of vector spaces, if F is equipped of its standard structure of vector space, where vector addition and scalar multiplication are done component by component. This isomorphism allows representing
SECTION 20
#17327808476553234-388: A free abelian group is a free abelian group, and, if G is a subgroup of a finitely generated free abelian group H (that is an abelian group that has a finite basis), then there is a basis e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} of H and an integer 0 ≤ k ≤ n such that
3381-987: A linear extension of f : S → Y {\displaystyle f:S\to Y} exists then the linear extension F : span S → Y {\displaystyle F:\operatorname {span} S\to Y} is unique and F ( c 1 s 1 + ⋯ c n s n ) = c 1 f ( s 1 ) + ⋯ + c n f ( s n ) {\displaystyle F\left(c_{1}s_{1}+\cdots c_{n}s_{n}\right)=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right)} holds for all n , c 1 , … , c n , {\displaystyle n,c_{1},\ldots ,c_{n},} and s 1 , … , s n {\displaystyle s_{1},\ldots ,s_{n}} as above. If S {\displaystyle S}
3528-988: A linear map F : span S → Y {\displaystyle F:\operatorname {span} S\to Y} if and only if whenever n > 0 {\displaystyle n>0} is an integer, c 1 , … , c n {\displaystyle c_{1},\ldots ,c_{n}} are scalars, and s 1 , … , s n ∈ S {\displaystyle s_{1},\ldots ,s_{n}\in S} are vectors such that 0 = c 1 s 1 + ⋯ + c n s n , {\displaystyle 0=c_{1}s_{1}+\cdots +c_{n}s_{n},} then necessarily 0 = c 1 f ( s 1 ) + ⋯ + c n f ( s n ) . {\displaystyle 0=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right).} If
3675-412: A linear map T : V → W , the image T ( V ) of V , and the inverse image T ( 0 ) of 0 (called kernel or null space), are linear subspaces of W and V , respectively. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums where v 1 , v 2 , ..., v k are in S , and a 1 , a 2 , ...,
3822-684: A linear map is one which preserves linear combinations . Denoting the zero elements of the vector spaces V {\displaystyle V} and W {\displaystyle W} by 0 V {\textstyle \mathbf {0} _{V}} and 0 W {\textstyle \mathbf {0} _{W}} respectively, it follows that f ( 0 V ) = 0 W . {\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.} Let c = 0 {\displaystyle c=0} and v ∈ V {\textstyle \mathbf {v} \in V} in
3969-521: A linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a conformal linear transformation . The composition of linear maps is linear: if f : V → W {\displaystyle f:V\to W} and g : W → Z {\textstyle g:W\to Z} are linear, then so is their composition g ∘ f : V → Z {\textstyle g\circ f:V\to Z} . It follows from this that
4116-917: A linear map on span { ( 1 , 0 ) , ( 0 , 1 ) } = R 2 . {\displaystyle \operatorname {span} \{(1,0),(0,1)\}=\mathbb {R} ^{2}.} The unique linear extension F : R 2 → R {\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} } is the map that sends ( x , y ) = x ( 1 , 0 ) + y ( 0 , 1 ) ∈ R 2 {\displaystyle (x,y)=x(1,0)+y(0,1)\in \mathbb {R} ^{2}} to F ( x , y ) = x ( − 1 ) + y ( 2 ) = − x + 2 y . {\displaystyle F(x,y)=x(-1)+y(2)=-x+2y.} Every (scalar-valued) linear functional f {\displaystyle f} defined on
4263-492: A lower dimension ); for example, it maps a plane through the origin in V {\displaystyle V} to either a plane through the origin in W {\displaystyle W} , a line through the origin in W {\displaystyle W} , or just the origin in W {\displaystyle W} . Linear maps can often be represented as matrices , and simple examples include rotation and reflection linear transformations . In
4410-505: A matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants". Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the work later. The telegraph required an explanatory system, and
4557-407: A one-dimensional vector space over itself is called a linear functional . These statements generalize to any left-module R M {\textstyle {}_{R}M} over a ring R {\displaystyle R} without modification, and to any right-module upon reversing of the scalar multiplication. Often, a linear map is constructed by defining it on a subset of
Linear algebra - Misplaced Pages Continue
4704-404: A part of the basis of W is mapped bijectively on a part of the basis of V , and that the remaining basis elements of W , if any, are mapped to zero. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results. A finite set of linear equations in a finite set of variables, for example, x 1 , x 2 , ..., x n , or x , y , ..., z
4851-408: A segment equipollent to pq . Other hypercomplex number systems also used the idea of a linear space with a basis . Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group . The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote
4998-655: A smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom. The index of an operator is precisely the Euler characteristic of the 2-term complex 0 → V → W → 0. In operator theory , the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem . No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on
5145-443: A subspace of V , then dim U ≤ dim V . In the case where V is finite-dimensional, the equality of the dimensions implies U = V . If U 1 and U 2 are subspaces of V , then where U 1 + U 2 denotes the span of U 1 ∪ U 2 . Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps . Their theory is thus an essential part of linear algebra. Let V be
5292-409: A system, one may associate its matrix and its right member vector Let T be the linear transformation associated to the matrix M . A solution of the system ( S ) is a vector Linear map In mathematics , and more specifically in linear algebra , a linear map (also called a linear mapping , linear transformation , vector space homomorphism , or in some contexts linear function )
5439-500: A vector ( a , b ), the value of a is the obstruction to there being a solution. An example illustrating the infinite-dimensional case is afforded by the map f : R → R , { a n } ↦ { b n } {\textstyle \left\{a_{n}\right\}\mapsto \left\{b_{n}\right\}} with b 1 = 0 and b n + 1 = a n for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of
5586-405: A vector by its inverse image under this isomorphism, that is by the coordinate vector ( a 1 , ..., a m ) or by the column matrix If W is another finite dimensional vector space (possibly the same), with a basis ( w 1 , ..., w n ) , a linear map f from W to V is well defined by its values on the basis elements, that is ( f ( w 1 ), ..., f ( w n )) . Thus, f
5733-399: A vector space V have the same cardinality , which is called the dimension of V ; this is the dimension theorem for vector spaces . Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. If any basis of V (and therefore every basis) has a finite number of elements, V is a finite-dimensional vector space . If U is
5880-424: A vector space and then extending by linearity to the linear span of the domain. Suppose X {\displaystyle X} and Y {\displaystyle Y} are vector spaces and f : S → Y {\displaystyle f:S\to Y} is a function defined on some subset S ⊆ X . {\displaystyle S\subseteq X.} Then
6027-528: A vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra . The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. Until
Linear algebra - Misplaced Pages Continue
6174-420: Is n + 1 {\displaystyle n+1} points in general linear position . A projective basis is n + 2 {\displaystyle n+2} points in general position, in a projective space of dimension n . A convex basis of a polytope is the set of the vertices of its convex hull . A cone basis consists of one point by edge of a polygonal cone. See also
6321-411: Is isomorphic to the general linear group GL ( n , K ) {\textstyle \operatorname {GL} (n,K)} of all n × n {\textstyle n\times n} invertible matrices with entries in K {\textstyle K} . If f : V → W {\textstyle f:V\to W} is linear, we define
6468-612: Is a function space , which is a common convention in functional analysis . Sometimes the term linear function has the same meaning as linear map , while in analysis it does not. A linear map from V {\displaystyle V} to W {\displaystyle W} always maps the origin of V {\displaystyle V} to the origin of W {\displaystyle W} . Moreover, it maps linear subspaces in V {\displaystyle V} onto linear subspaces in W {\displaystyle W} (possibly of
6615-572: Is a linear isomorphism from the vector space F n {\displaystyle F^{n}} onto V . In other words, F n {\displaystyle F^{n}} is the coordinate space of V , and the n -tuple φ − 1 ( v ) {\displaystyle \varphi ^{-1}(\mathbf {v} )} is the coordinate vector of v . The inverse image by φ {\displaystyle \varphi } of b i {\displaystyle \mathbf {b} _{i}}
6762-410: Is a mapping V → W {\displaystyle V\to W} between two vector spaces that preserves the operations of vector addition and scalar multiplication . The same names and the same definition are also used for the more general case of modules over a ring ; see Module homomorphism . If a linear map is a bijection then it is called a linear isomorphism . In
6909-457: Is a set V equipped with two binary operations . Elements of V are called vectors , and elements of F are called scalars . The first operation, vector addition , takes any two vectors v and w and outputs a third vector v + w . The second operation, scalar multiplication , takes any scalar a and any vector v and outputs a new vector a v . The axioms that addition and scalar multiplication must satisfy are
7056-698: Is a subspace of V {\textstyle V} and im ( f ) {\textstyle \operatorname {im} (f)} is a subspace of W {\textstyle W} . The following dimension formula is known as the rank–nullity theorem : dim ( ker ( f ) ) + dim ( im ( f ) ) = dim ( V ) . {\displaystyle \dim(\ker(f))+\dim(\operatorname {im} (f))=\dim(V).} The number dim ( im ( f ) ) {\textstyle \dim(\operatorname {im} (f))}
7203-413: Is a basis of V . Since L max belongs to X , we already know that L max is a linearly independent subset of V . If there were some vector w of V that is not in the span of L max , then w would not be an element of L max either. Let L w = L max ∪ { w } . This set is an element of X , that is, it is a linearly independent subset of V (because w
7350-510: Is a linear map, f ( v ) = f ( c 1 v 1 + ⋯ + c n v n ) = c 1 f ( v 1 ) + ⋯ + c n f ( v n ) , {\displaystyle f(\mathbf {v} )=f(c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n})=c_{1}f(\mathbf {v} _{1})+\cdots +c_{n}f\left(\mathbf {v} _{n}\right),} which implies that
7497-404: Is a linear map. In particular, if f {\displaystyle f} has a linear extension to span S , {\displaystyle \operatorname {span} S,} then it has a linear extension to all of X . {\displaystyle X.} The map f : S → Y {\displaystyle f:S\to Y} can be extended to
SECTION 50
#17327808476557644-399: Is a linearly independent subset of V that spans V . This means that a subset B of V is a basis if it satisfies the two following conditions: The scalars a i {\displaystyle a_{i}} are called the coordinates of the vector v with respect to the basis B , and by the first property they are uniquely determined. A vector space that has
7791-400: Is a manifestation of the so-called measure concentration phenomenon . The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the n -dimensional cube [−1, 1] as a function of dimension, n . A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If
7938-448: Is a vector ( a 1 j ⋮ a m j ) {\displaystyle {\begin{pmatrix}a_{1j}\\\vdots \\a_{mj}\end{pmatrix}}} corresponding to f ( v j ) {\displaystyle f(\mathbf {v} _{j})} as defined above. To define it more clearly, for some column j {\displaystyle j} that corresponds to
8085-734: Is also called a coordinate frame or simply a frame (for example, a Cartesian frame or an affine frame ). Let, as usual, F n {\displaystyle F^{n}} be the set of the n -tuples of elements of F . This set is an F -vector space, with addition and scalar multiplication defined component-wise. The map φ : ( λ 1 , … , λ n ) ↦ λ 1 b 1 + ⋯ + λ n b n {\displaystyle \varphi :(\lambda _{1},\ldots ,\lambda _{n})\mapsto \lambda _{1}\mathbf {b} _{1}+\cdots +\lambda _{n}\mathbf {b} _{n}}
8232-392: Is also called the rank of f {\textstyle f} and written as rank ( f ) {\textstyle \operatorname {rank} (f)} , or sometimes, ρ ( f ) {\textstyle \rho (f)} ; the number dim ( ker ( f ) ) {\textstyle \dim(\ker(f))}
8379-454: Is also linear. Thus the set L ( V , W ) {\textstyle {\mathcal {L}}(V,W)} of linear maps from V {\textstyle V} to W {\textstyle W} itself forms a vector space over K {\textstyle K} , sometimes denoted Hom ( V , W ) {\textstyle \operatorname {Hom} (V,W)} . Furthermore, in
8526-406: Is an endomorphism of V {\textstyle V} ; the set of all such endomorphisms End ( V ) {\textstyle \operatorname {End} (V)} together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K {\textstyle K} (and in particular
8673-521: Is an element of X . Therefore, L Y is an upper bound for Y in ( X , ⊆) : it is an element of X , that contains every element of Y . As X is nonempty, and every totally ordered subset of ( X , ⊆) has an upper bound in X , Zorn's lemma asserts that X has a maximal element. In other words, there exists some element L max of X satisfying the condition that whenever L max ⊆ L for some element L of X , then L = L max . It remains to prove that L max
8820-529: Is any real number. A simple basis of this vector space consists of the two vectors e 1 = (1, 0) and e 2 = (0, 1) . These vectors form a basis (called the standard basis ) because any vector v = ( a , b ) of R may be uniquely written as v = a e 1 + b e 2 . {\displaystyle \mathbf {v} =a\mathbf {e} _{1}+b\mathbf {e} _{2}.} Any other pair of linearly independent vectors of R , such as (1, 1) and (−1, 2) , forms also
8967-560: Is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication. By the associativity of the addition operation denoted as +, for any vectors u 1 , … , u n ∈ V {\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V} and scalars c 1 , … , c n ∈ K , {\textstyle c_{1},\ldots ,c_{n}\in K,}
SECTION 60
#17327808476559114-420: Is called a system of linear equations or a linear system . Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems. For example, let be a linear system. To such
9261-438: Is called the nullity of f {\textstyle f} and written as null ( f ) {\textstyle \operatorname {null} (f)} or ν ( f ) {\textstyle \nu (f)} . If V {\textstyle V} and W {\textstyle W} are finite-dimensional, bases have been chosen and f {\textstyle f}
9408-469: Is customary to refer to B o l d {\displaystyle B_{\mathrm {old} }} and B n e w {\displaystyle B_{\mathrm {new} }} as the old basis and the new basis , respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has expressions involving the old coordinates, and if one wants to obtain equivalent expressions in terms of
9555-414: Is defined as coker ( f ) := W / f ( V ) = W / im ( f ) . {\displaystyle \operatorname {coker} (f):=W/f(V)=W/\operatorname {im} (f).} This is the dual notion to the kernel: just as the kernel is a sub space of the domain, the co-kernel is a quotient space of the target. Formally, one has
9702-421: Is denoted, as usual, by ⊆ . Let Y be a subset of X that is totally ordered by ⊆ , and let L Y be the union of all the elements of Y (which are themselves certain subsets of V ). Since ( Y , ⊆) is totally ordered, every finite subset of L Y is a subset of an element of Y , which is a linearly independent subset of V , and hence L Y is linearly independent. Thus L Y
9849-732: Is equal to 1, is a countable Hamel basis. In the study of Fourier series , one learns that the functions {1} ∪ { sin( nx ), cos( nx ) : n = 1, 2, 3, ... } are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions f satisfying ∫ 0 2 π | f ( x ) | 2 d x < ∞ . {\displaystyle \int _{0}^{2\pi }\left|f(x)\right|^{2}\,dx<\infty .} The functions {1} ∪ { sin( nx ), cos( nx ) : n = 1, 2, 3, ... } are linearly independent, and every function f that
9996-540: Is equivalent to T being both one-to-one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism . If T : V → V is an endomorphism, then: Basis (linear algebra) In mathematics , a set B of vectors in a vector space V is called a basis ( pl. : bases ) if every element of V may be written in a unique way as a finite linear combination of elements of B . The coefficients of this linear combination are referred to as components or coordinates of
10143-535: Is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials ) is also a basis. (Such a set of polynomials is called a polynomial sequence .) But there are also many bases for F [ X ] that are not of this form. Many properties of finite bases result from the Steinitz exchange lemma , which states that, for any vector space V , given a finite spanning set S and
10290-444: Is given by polynomial rings . If F is a field, the collection F [ X ] of all polynomials in one indeterminate X with coefficients in F is an F -vector space. One basis for this space is the monomial basis B , consisting of all monomials : B = { 1 , X , X 2 , … } . {\displaystyle B=\{1,X,X^{2},\ldots \}.} Any set of polynomials such that there
10437-789: Is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If X is an infinite-dimensional normed vector space that is complete (i.e. X is a Banach space ), then any Hamel basis of X is necessarily uncountable . This is a consequence of the Baire category theorem . The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional ( non-complete ) normed spaces that have countable Hamel bases. Consider c 00 {\displaystyle c_{00}} ,
10584-422: Is linear and α {\textstyle \alpha } is an element of the ground field K {\textstyle K} , then the map α f {\textstyle \alpha f} , defined by ( α f ) ( x ) = α ( f ( x ) ) {\textstyle (\alpha f)(\mathbf {x} )=\alpha (f(\mathbf {x} ))} ,
10731-521: Is linearly independent then every function f : S → Y {\displaystyle f:S\to Y} into any vector space has a linear extension to a (linear) map span S → Y {\displaystyle \;\operatorname {span} S\to Y} (the converse is also true). For example, if X = R 2 {\displaystyle X=\mathbb {R} ^{2}} and Y = R {\displaystyle Y=\mathbb {R} } then
10878-442: Is not in the span of L max , and L max is independent). As L max ⊆ L w , and L max ≠ L w (because L w contains the vector w that is not contained in L max ), this contradicts the maximality of L max . Thus this shows that L max spans V . Hence L max is linearly independent and spans V . It is thus a basis of V , and this proves that every vector space has
11025-439: Is often useful to express the coordinates of a vector x with respect to B o l d {\displaystyle B_{\mathrm {old} }} in terms of the coordinates with respect to B n e w . {\displaystyle B_{\mathrm {new} }.} This can be done by the change-of-basis formula , that is described below. The subscripts "old" and "new" have been chosen because it
11172-404: Is represented by the matrix A {\textstyle A} , then the rank and nullity of f {\textstyle f} are equal to the rank and nullity of the matrix A {\textstyle A} , respectively. A subtler invariant of a linear transformation f : V → W {\textstyle f:V\to W} is the co kernel , which
11319-451: Is said to be a linear map if for any two vectors u , v ∈ V {\textstyle \mathbf {u} ,\mathbf {v} \in V} and any scalar c ∈ K {\displaystyle c\in K} the following two conditions are satisfied: Thus, a linear map is said to be operation preserving . In other words, it does not matter whether the linear map
11466-743: Is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that lim n → ∞ ∫ 0 2 π | a 0 + ∑ k = 1 n ( a k cos ( k x ) + b k sin ( k x ) ) − f ( x ) | 2 d x = 0 {\displaystyle \lim _{n\to \infty }\int _{0}^{2\pi }{\biggl |}a_{0}+\sum _{k=1}^{n}\left(a_{k}\cos \left(kx\right)+b_{k}\sin \left(kx\right)\right)-f(x){\biggr |}^{2}dx=0} for suitable (real or complex) coefficients
11613-513: Is that not every module has a basis. A module that has a basis is called a free module . Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions . A module over the integers is exactly the same thing as an abelian group . Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of
11760-404: Is the n -tuple e i {\displaystyle \mathbf {e} _{i}} all of whose components are 0, except the i th that is 1. The e i {\displaystyle \mathbf {e} _{i}} form an ordered basis of F n {\displaystyle F^{n}} , which is called its standard basis or canonical basis . The ordered basis B
11907-599: Is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1. For a linear operator with finite-dimensional kernel and co-kernel, one may define index as: ind ( f ) := dim ( ker ( f ) ) − dim ( coker ( f ) ) , {\displaystyle \operatorname {ind} (f):=\dim(\ker(f))-\dim(\operatorname {coker} (f)),} namely
12054-595: Is the group of units in the ring End ( V ) {\textstyle \operatorname {End} (V)} . If V {\textstyle V} has finite dimension n {\textstyle n} , then End ( V ) {\textstyle \operatorname {End} (V)} is isomorphic to the associative algebra of all n × n {\textstyle n\times n} matrices with entries in K {\textstyle K} . The automorphism group of V {\textstyle V}
12201-481: Is the image by φ {\displaystyle \varphi } of the canonical basis of F n {\displaystyle F^{n}} . It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of F n {\displaystyle F^{n}} , and that every linear isomorphism from F n {\displaystyle F^{n}} onto V may be defined as
12348-463: Is the matrix of f {\displaystyle f} . In other words, every column j = 1 , … , n {\displaystyle j=1,\ldots ,n} has a corresponding vector f ( v j ) {\displaystyle f(\mathbf {v} _{j})} whose coordinates a 1 j , ⋯ , a m j {\displaystyle a_{1j},\cdots ,a_{mj}} are
12495-533: Is the smallest infinite cardinal, the cardinal of the integers. The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for topological vector spaces – a large class of vector spaces including e.g. Hilbert spaces , Banach spaces , or Fréchet spaces . The preference of other types of bases for infinite-dimensional spaces
12642-474: Is their pointwise sum f 1 + f 2 {\displaystyle f_{1}+f_{2}} , which is defined by ( f 1 + f 2 ) ( x ) = f 1 ( x ) + f 2 ( x ) {\displaystyle (f_{1}+f_{2})(\mathbf {x} )=f_{1}(\mathbf {x} )+f_{2}(\mathbf {x} )} . If f : V → W {\textstyle f:V\to W}
12789-415: Is well represented by the list of the corresponding column matrices. That is, if for j = 1, ..., n , then f is represented by the matrix with m rows and n columns. Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing
12936-516: Is ε-orthogonal to y if | ⟨ x , y ⟩ | / ( ‖ x ‖ ‖ y ‖ ) < ε {\displaystyle \left|\left\langle x,y\right\rangle \right|/\left(\left\|x\right\|\left\|y\right\|\right)<\varepsilon } (that is, cosine of the angle between x and y is less than ε ). In high dimensions, two independent random vectors are with high probability almost orthogonal, and
13083-431: The axiom of choice or a weaker form of it, such as the ultrafilter lemma . If V is a vector space over a field F , then: If V is a vector space of dimension n , then: Let V be a vector space of finite dimension n over a field F , and B = { b 1 , … , b n } {\displaystyle B=\{\mathbf {b} _{1},\ldots ,\mathbf {b} _{n}\}} be
13230-411: The class of all vector spaces over a given field K , together with K -linear maps as morphisms , forms a category . The inverse of a linear map, when defined, is again a linear map. If f 1 : V → W {\textstyle f_{1}:V\to W} and f 2 : V → W {\textstyle f_{2}:V\to W} are linear, then so
13377-725: The column vectors of the coordinates of v in the old and the new basis respectively, then the formula for changing coordinates is X = A Y . {\displaystyle X=AY.} The formula can be proven by considering the decomposition of the vector x on the two bases: one has x = ∑ i = 1 n x i v i , {\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {v} _{i},} and x = ∑ j = 1 n y j w j = ∑ j = 1 n y j ∑ i = 1 n
13524-513: The coordinates of v over B . However, if one talks of the set of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same set of coefficients. For example, 3 b 1 + 2 b 2 {\displaystyle 3\mathbf {b} _{1}+2\mathbf {b} _{2}} and 2 b 1 + 3 b 2 {\displaystyle 2\mathbf {b} _{1}+3\mathbf {b} _{2}} have
13671-419: The dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications in the study of crystal structures and frames of reference . A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C )
13818-410: The exact sequence 0 → ker ( f ) → V → W → coker ( f ) → 0. {\displaystyle 0\to \ker(f)\to V\to W\to \operatorname {coker} (f)\to 0.} These can be interpreted thus: given a linear equation f ( v ) = w to solve, The dimension of the co-kernel and the dimension of
13965-882: The kernel and the image or range of f {\textstyle f} by ker ( f ) = { x ∈ V : f ( x ) = 0 } im ( f ) = { w ∈ W : w = f ( x ) , x ∈ V } {\displaystyle {\begin{aligned}\ker(f)&=\{\,\mathbf {x} \in V:f(\mathbf {x} )=\mathbf {0} \,\}\\\operatorname {im} (f)&=\{\,\mathbf {w} \in W:\mathbf {w} =f(\mathbf {x} ),\mathbf {x} \in V\,\}\end{aligned}}} ker ( f ) {\textstyle \ker(f)}
14112-452: The n -tuple with all components equal to 0, except the i th, which is 1. Then e 1 , … , e n {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{n}} is a basis of F n , {\displaystyle F^{n},} which is called the standard basis of F n . {\displaystyle F^{n}.} A different flavor of example
14259-529: The ordered pairs of real numbers is a vector space under the operations of component-wise addition ( a , b ) + ( c , d ) = ( a + c , b + d ) {\displaystyle (a,b)+(c,d)=(a+c,b+d)} and scalar multiplication λ ( a , b ) = ( λ a , λ b ) , {\displaystyle \lambda (a,b)=(\lambda a,\lambda b),} where λ {\displaystyle \lambda }
14406-703: The "longer" method going clockwise from the same point such that [ v ] B ′ {\textstyle \left[\mathbf {v} \right]_{B'}} is left-multiplied with P − 1 A P {\textstyle P^{-1}AP} , or P − 1 A P [ v ] B ′ = [ T ( v ) ] B ′ {\textstyle P^{-1}AP\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}} . In two- dimensional space R linear maps are described by 2 × 2 matrices . These are some examples: If
14553-527: The 1873 publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression. Linear algebra is flat differential geometry and serves in tangent spaces to manifolds . Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations , and much of the history of linear algebra is the history of Lorentz transformations . The first modern and more precise definition of
14700-406: The 19th century, linear algebra was introduced through systems of linear equations and matrices . In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic , more general (not limited to the finite-dimensional case), and conceptually simpler, although more abstract. A vector space over a field F (often the field of the real numbers )
14847-435: The angle between the vectors was within π/2 ± 0.037π/2 then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within π/2 ± 0.037π/2 then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of
14994-404: The application of linear algebra to function spaces . Linear algebra is also used in most sciences and fields of engineering , because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems , which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations , using the fact that the differential of
15141-425: The assignment ( 1 , 0 ) → − 1 {\displaystyle (1,0)\to -1} and ( 0 , 1 ) → 2 {\displaystyle (0,1)\to 2} can be linearly extended from the linearly independent set of vectors S := { ( 1 , 0 ) , ( 0 , 1 ) } {\displaystyle S:=\{(1,0),(0,1)\}} to
15288-448: The basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering the basis elements. In order to emphasize that an order has been chosen, one speaks of an ordered basis , which is therefore not simply an unstructured set , but a sequence , an indexed family , or similar; see § Ordered bases and coordinates below. The set R of
15435-512: The bottom right corner [ T ( v ) ] B ′ {\textstyle \left[T\left(\mathbf {v} \right)\right]_{B'}} , one would left-multiply—that is, A ′ [ v ] B ′ = [ T ( v ) ] B ′ {\textstyle A'\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}} . The equivalent method would be
15582-411: The case of the real numbers R viewed as a vector space over the field Q of rational numbers, Hamel bases are uncountable, and have specifically the cardinality of the continuum, which is the cardinal number 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} , where ℵ 0 {\displaystyle \aleph _{0}} ( aleph-nought )
15729-424: The case that V = W {\textstyle V=W} , this vector space, denoted End ( V ) {\textstyle \operatorname {End} (V)} , is an associative algebra under composition of maps , since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below. Given again
15876-595: The case where V = W {\displaystyle V=W} , a linear map is called a linear endomorphism . Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V {\displaystyle V} and W {\displaystyle W} are real vector spaces (not necessarily with V = W {\displaystyle V=W} ), or it can be used to emphasize that V {\displaystyle V}
16023-435: The chain) is recorded. For each n , 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented. Let V be any vector space over some field F . Let X be the set of all linearly independent subsets of V . The set X is nonempty since the empty set is an independent subset of V , and it is partially ordered by inclusion, which
16170-497: The classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the co-kernel ( ℵ 0 + 0 = ℵ 0 + 1 {\textstyle \aleph _{0}+0=\aleph _{0}+1} ), but in
16317-498: The codomain of f . {\displaystyle f.} When the subset S {\displaystyle S} is a vector subspace of X {\displaystyle X} then a ( Y {\displaystyle Y} -valued) linear extension of f {\displaystyle f} to all of X {\displaystyle X} is guaranteed to exist if (and only if) f : S → Y {\displaystyle f:S\to Y}
16464-515: The context of infinite-dimensional vector spaces over the real or complex numbers, the term Hamel basis (named after Georg Hamel ) or algebraic basis can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are orthogonal bases on Hilbert spaces , Schauder bases , and Markushevich bases on normed linear spaces . In
16611-400: The coordinates of a vector x over the old and the new basis respectively, the change-of-basis formula is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . This formula may be concisely written in matrix notation. Let A be
16758-424: The definition of a vector space by a ring , one gets the definition of a module . For modules, linear independence and spanning sets are defined exactly as for vector spaces, although " generating set " is more commonly used than that of "spanning set". Like for vector spaces, a basis of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces
16905-428: The degrees of freedom minus the number of constraints. For a transformation between finite-dimensional vector spaces, this is just the difference dim( V ) − dim( W ), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from
17052-556: The domain of f {\displaystyle f} ) then there exists a linear extension to X {\displaystyle X} that is also dominated by p . {\displaystyle p.} If V {\displaystyle V} and W {\displaystyle W} are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from V {\displaystyle V} to W {\displaystyle W} can be represented by
17199-462: The elements of column j {\displaystyle j} . A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen. The matrices of a linear transformation can be represented visually: Such that starting in the bottom left corner [ v ] B ′ {\textstyle \left[\mathbf {v} \right]_{B'}} and looking for
17346-416: The equation det[ x 1 ⋯ x n ] = 0 (zero determinant of the matrix with columns x i ), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases. It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product , x
17493-433: The equation for homogeneity of degree 1: f ( 0 V ) = f ( 0 v ) = 0 f ( v ) = 0 W . {\displaystyle f(\mathbf {0} _{V})=f(0\mathbf {v} )=0f(\mathbf {v} )=\mathbf {0} _{W}.} A linear map V → K {\displaystyle V\to K} with K {\displaystyle K} viewed as
17640-408: The finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication , the addition of linear maps corresponds to the matrix addition , and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars. A linear transformation f : V → V {\textstyle f:V\to V}
17787-450: The following equality holds: f ( c 1 u 1 + ⋯ + c n u n ) = c 1 f ( u 1 ) + ⋯ + c n f ( u n ) . {\displaystyle f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).} Thus
17934-548: The following. (In the list below, u , v and w are arbitrary elements of V , and a and b are arbitrary scalars in the field F .) The first four axioms mean that V is an abelian group under addition. An element of a specific vector space may have various nature; for example, it could be a sequence , a function , a polynomial or a matrix . Linear algebra is concerned with those properties of such objects that are common to all vector spaces. Linear maps are mappings between vector spaces that preserve
18081-655: The function f {\displaystyle f} is entirely determined by the values of a i j {\displaystyle a_{ij}} . If we put these values into an m × n {\displaystyle m\times n} matrix M {\displaystyle M} , then we can conveniently use it to compute the vector output of f {\displaystyle f} for any vector in V {\displaystyle V} . To get M {\displaystyle M} , every column j {\displaystyle j} of M {\displaystyle M}
18228-412: The function f is entirely determined by the vectors f ( v 1 ) , … , f ( v n ) {\displaystyle f(\mathbf {v} _{1}),\ldots ,f(\mathbf {v} _{n})} . Now let { w 1 , … , w m } {\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}} be
18375-398: The image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space W / f ( V ) is the dimension of the target space minus the dimension of the image. As a simple example, consider the map f : R → R , given by f ( x , y ) = (0, y ). Then for an equation f ( x , y ) = ( a , b ) to have a solution, we must have
18522-417: The induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces . More precisely, a linear subspace of a vector space V over a field F is a subset W of V such that u + v and a u are in W , for every u , v in W , and every a in F . (These conditions suffice for implying that W is a vector space.) For example, given
18669-403: The infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map h : R → R , { a n } ↦ { c n } {\textstyle \left\{a_{n}\right\}\mapsto \left\{c_{n}\right\}} with c n = a n + 1 . Its image
18816-548: The introduction in 1637 by René Descartes of coordinates in geometry . In fact, in this new geometry, now called Cartesian geometry , lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations. The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule . Later, Gauss further described
18963-826: The isomorphism that maps the canonical basis of F n {\displaystyle F^{n}} onto a given ordered basis of V . In other words, it is equivalent to define an ordered basis of V , or a linear isomorphism from F n {\displaystyle F^{n}} onto V . Let V be a vector space of dimension n over a field F . Given two (ordered) bases B old = ( v 1 , … , v n ) {\displaystyle B_{\text{old}}=(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})} and B new = ( w 1 , … , w n ) {\displaystyle B_{\text{new}}=(\mathbf {w} _{1},\ldots ,\mathbf {w} _{n})} of V , it
19110-421: The language of category theory , linear maps are the morphisms of vector spaces, and they form a category equivalent to the one of matrices . Let V {\displaystyle V} and W {\displaystyle W} be vector spaces over the same field K {\displaystyle K} . A function f : V → W {\displaystyle f:V\to W}
19257-486: The mapping f ( v j ) {\displaystyle f(\mathbf {v} _{j})} , M = ( ⋯ a 1 j ⋯ ⋮ a m j ) {\displaystyle \mathbf {M} ={\begin{pmatrix}\ \cdots &a_{1j}&\cdots \ \\&\vdots &\\&a_{mj}&\end{pmatrix}}} where M {\displaystyle M}
19404-495: The matrix of the a i , j {\displaystyle a_{i,j}} , and X = [ x 1 ⋮ x n ] and Y = [ y 1 ⋮ y n ] {\displaystyle X={\begin{bmatrix}x_{1}\\\vdots \\x_{n}\end{bmatrix}}\quad {\text{and}}\quad Y={\begin{bmatrix}y_{1}\\\vdots \\y_{n}\end{bmatrix}}} be
19551-491: The method of elimination, which was initially listed as an advancement in geodesy . In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix , which is Latin for womb . Linear algebra grew with ideas noted in the complex plane . For instance, two numbers w and z in C {\displaystyle \mathbb {C} } have
19698-713: The new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates. Typically, the new basis vectors are given by their coordinates over the old basis, that is, w j = ∑ i = 1 n a i , j v i . {\displaystyle \mathbf {w} _{j}=\sum _{i=1}^{n}a_{i,j}\mathbf {v} _{i}.} If ( x 1 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} and ( y 1 , … , y n ) {\displaystyle (y_{1},\ldots ,y_{n})} are
19845-621: The number of independent random vectors, which all are with given high probability pairwise almost orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in n -dimensional ball. Choose N independent random vectors from a ball (they are independent and identically distributed ). Let θ be a small positive number. Then for N random vectors are all pairwise ε-orthogonal with probability 1 − θ . This N growth exponentially with dimension n and N ≫ n {\displaystyle N\gg n} for sufficiently big n . This property of random bases
19992-431: The other by elementary row and column operations . For a matrix representing a linear map from W to V , the row operations correspond to change of bases in V and the column operations correspond to change of bases in W . Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector spaces, this means that, for any linear map from W to V , there are bases such that
20139-424: The result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts. Two matrices that encode the same linear transformation in different bases are called similar . It can be proved that two matrices are similar if and only if one can transform one into
20286-425: The same set of coefficients {2, 3} , and are different. It is therefore often convenient to work with an ordered basis ; this is typically done by indexing the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis, especially when used in conjunction with an origin ,
20433-416: The same vector space, a linear map T : V → V is also known as a linear operator on V . A bijective linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an isomorphism . Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in
20580-524: The sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm . The study of those subsets of vector spaces that are in themselves vector spaces under
20727-563: The set of all automorphisms of V {\textstyle V} forms a group , the automorphism group of V {\textstyle V} which is denoted by Aut ( V ) {\textstyle \operatorname {Aut} (V)} or GL ( V ) {\textstyle \operatorname {GL} (V)} . Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut ( V ) {\textstyle \operatorname {Aut} (V)}
20874-436: The space of the sequences x = ( x n ) {\displaystyle x=(x_{n})} of real numbers that have only finitely many non-zero elements, with the norm ‖ x ‖ = sup n | x n | {\textstyle \|x\|=\sup _{n}|x_{n}|} . Its standard basis , consisting of the sequences having only one non-zero element, which
21021-408: The uniqueness of the decomposition of a vector over a basis, here B old {\displaystyle B_{\text{old}}} ; that is x i = ∑ j = 1 n a i , j y j , {\displaystyle x_{i}=\sum _{j=1}^{n}a_{i,j}y_{j},} for i = 1, ..., n . If one replaces the field occurring in
21168-426: The vector space. Let V and W denote vector spaces over a field F and let T : V → W be a linear map. T is said to be injective or a monomorphism if any of the following equivalent conditions are true: T is said to be surjective or an epimorphism if any of the following equivalent conditions are true: T is said to be an isomorphism if it is both left- and right-invertible. This
21315-409: The vector with respect to B . The elements of a basis are called basis vectors . Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B . In other words, a basis is a linearly independent spanning set . A vector space can have several bases; however all the bases have the same number of elements, called
21462-432: The vector-space structure. Given two vector spaces V and W over a field F , a linear map (also called, in some contexts, linear transformation or linear mapping) is a map that is compatible with addition and scalar multiplication, that is for any vectors u , v in V and scalar a in F . This implies that for any vectors u , v in V and scalars a , b in F , one has When V = W are
21609-487: The zero vector as a linear combination of elements of S is to take zero for every coefficient a i . A set of vectors that spans a vector space is called a spanning set or generating set . If a spanning set S is linearly dependent (that is not linearly independent), then some element w of S is in the span of the other elements of S , and the span would remain the same if one were to remove w from S . One may continue to remove elements of S until getting
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