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67-694: The Canada Basin is a deep oceanic basin within the Arctic Ocean . It is part of the Amerasian Basin and lies off the coast of Alaska and northwest Canada between the Chukchi Plateau north of Alaska and the Alpha Ridge north of Ellesmere Island . This Arctic -related article is a stub . You can help Misplaced Pages by expanding it . Oceanic basin Most commonly the ocean

134-398: A 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] = ( a 11

201-830: A 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ) . {\displaystyle \mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{bmatrix}}={\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}.} This may be abbreviated by writing only

268-401: A i , j ) 1 ≤ i , j ≤ n {\displaystyle \mathbf {A} =(a_{i,j})_{1\leq i,j\leq n}} in the case that n = m {\displaystyle n=m} . Matrices are usually symbolized using upper-case letters (such as A {\displaystyle {\mathbf {A} }} in the examples above), while

335-672: A field or a ring . In this section, it is supposed that matrix entries belong to a fixed ring, which is typically a field of numbers. The sum A + B of two m × n matrices A and B is calculated entrywise: ( A + B ) i , j = A i , j + B i , j , 1 ≤ i ≤ m , 1 ≤ j ≤ n . {\displaystyle ({\mathbf {A}}+{\mathbf {B}})_{i,j}={\mathbf {A}}_{i,j}+{\mathbf {B}}_{i,j},\quad 1\leq i\leq m,\quad 1\leq j\leq n.} For example, The product c A of

402-700: A k -by- m matrix B represents another linear map g : R m → R k {\displaystyle g:\mathbb {R} ^{m}\to \mathbb {R} ^{k}} , then the composition g ∘ f is represented by BA since ( g ∘ f ) ( x ) = g ( f ( x ) ) = g ( A x ) = B ( A x ) = ( B A ) x . {\displaystyle (g\circ f)({\mathbf {x}})=g(f({\mathbf {x}}))=g({\mathbf {Ax}})={\mathbf {B}}({\mathbf {Ax}})=({\mathbf {BA}}){\mathbf {x}}.} The last equality follows from

469-467: A matrix ( pl. : matrices ) is a rectangular array or table of numbers , symbols , or expressions , with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, [ 1 9 − 13 20 5 − 6 ] {\displaystyle {\begin{bmatrix}1&9&-13\\20&5&-6\end{bmatrix}}}

536-614: A , b ) , ( a + c , b + d ) , and ( c , d ) . The parallelogram pictured at the right is obtained by multiplying A with each of the column vectors [ 0 0 ] , [ 1 0 ] , [ 1 1 ] {\displaystyle {\begin{bmatrix}0\\0\end{bmatrix}},{\begin{bmatrix}1\\0\end{bmatrix}},{\begin{bmatrix}1\\1\end{bmatrix}}} , and [ 0 1 ] {\displaystyle {\begin{bmatrix}0\\1\end{bmatrix}}} in turn. These vectors define

603-406: A 2-by-3 submatrix by removing row 3 and column 2: The minors and cofactors of a matrix are found by computing the determinant of certain submatrices. A principal submatrix is a square submatrix obtained by removing certain rows and columns. The definition varies from author to author. According to some authors, a principal submatrix is a submatrix in which the set of row indices that remain

670-469: A global ocean model. These trajectories are of particles that move only on the surface of the ocean. The model outcome gives the probability of a particle at a certain grid point to end up somewhere else on the ocean's surface. With the model outcome a matrix can be created from which the Eigenvectors and Eigenvalues are taken. These Eigenvectors show regions of attraction, aka regions where things on

737-407: A matrix are called rows and columns , respectively. The size of a matrix is defined by the number of rows and columns it contains. There is no limit to the number of rows and columns, that a matrix (in the usual sense) can have as long as they are positive integers. A matrix with m {\displaystyle {m}} rows and n {\displaystyle {n}} columns

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804-429: A matrix over a field F is a rectangular array of elements of F . A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers . More general types of entries are discussed below . For instance, this is a real matrix: The numbers, symbols, or expressions in the matrix are called its entries or its elements . The horizontal and vertical lines of entries in

871-412: A matrix plus the rank equals the number of columns of the matrix. A square matrix is a matrix with the same number of rows and columns. An n -by- n matrix is known as a square matrix of order n . Any two square matrices of the same order can be added and multiplied. The entries a ii form the main diagonal of a square matrix. They lie on the imaginary line that runs from the top left corner to

938-596: A nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant. In geometry , matrices are widely used for specifying and representing geometric transformations (for example rotations ) and coordinate changes . In numerical analysis , many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly, or through their use in geometry and numerical analysis. Matrix theory

1005-464: A number c (also called a scalar in this context) and a matrix A is computed by multiplying every entry of A by c : ( c A ) i , j = c ⋅ A i , j {\displaystyle (c{\mathbf {A}})_{i,j}=c\cdot {\mathbf {A}}_{i,j}} This operation is called scalar multiplication , but its result is not named "scalar product" to avoid confusion, since "scalar product"

1072-504: A particle on the ocean surface in a certain region is more likely to stay in the same region than to pass over to a different one. Depending on the chemical composition and the physical state, the Earth can be divided into three major components:  the mantle , the core , and the crust . The crust is referred to as the outside layer of the Earth. It is made of solid rock, mostly basalt and granite . The crust that lies below sea level

1139-507: A single generic term, possibly along with indices, as in A = ( a i j ) , [ a i j ] , or ( a i j ) 1 ≤ i ≤ m , 1 ≤ j ≤ n {\displaystyle \mathbf {A} =\left(a_{ij}\right),\quad \left[a_{ij}\right],\quad {\text{or}}\quad \left(a_{ij}\right)_{1\leq i\leq m,\;1\leq j\leq n}} or A = (

1206-743: A subscript. For instance, the matrix A {\displaystyle \mathbf {A} } above is 3 × 4 {\displaystyle 3\times 4} , and can be defined as A = [ i − j ] ( i = 1 , 2 , 3 ; j = 1 , … , 4 ) {\displaystyle {\mathbf {A} }=[i-j](i=1,2,3;j=1,\dots ,4)} or A = [ i − j ] 3 × 4 {\displaystyle {\mathbf {A} }=[i-j]_{3\times 4}} . Some programming languages utilize doubly subscripted arrays (or arrays of arrays) to represent an m -by- n matrix. Some programming languages start

1273-399: Is commutative , that is, the matrix sum does not depend on the order of the summands: A + B = B + A . The transpose is compatible with addition and scalar multiplication, as expressed by ( c A ) = c ( A ) and ( A + B ) = A + B . Finally, ( A ) = A . Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as

1340-446: Is a 3 × 2 {\displaystyle {3\times 2}} matrix. Matrices with a single row are called row vectors , and those with a single column are called column vectors . A matrix with the same number of rows and columns is called a square matrix . A matrix with an infinite number of rows or columns (or both) is called an infinite matrix . In some contexts, such as computer algebra programs , it

1407-619: Is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " 2 × 3 {\displaystyle 2\times 3} matrix", or a matrix of dimension 2 × 3 {\displaystyle 2\times 3} . Matrices are commonly related to linear algebra . Notable exceptions include incidence matrices and adjacency matrices in graph theory . This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such. Square matrices , matrices with

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1474-555: Is affected not only by the volume of the ocean basin, but also by the volume of water in them. Factors that influence the volume of the ocean basins are: The Atlantic Ocean and the Arctic Ocean are good examples of active, growing oceanic basins, whereas the Mediterranean Sea is shrinking. The Pacific Ocean is also an active, shrinking oceanic basin, even though it has both spreading ridge and oceanic trenches. Perhaps

1541-470: Is an m × n matrix, x designates a column vector (that is, n ×1 -matrix) of n variables x 1 , x 2 , ..., x n , and b is an m ×1 -column vector, then the matrix equation is equivalent to the system of linear equations Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If n = m and the equations are independent , then this can be done by writing where A

1608-456: Is called an m × n {\displaystyle {m\times n}} matrix, or m {\displaystyle {m}} -by- n {\displaystyle {n}} matrix, where m {\displaystyle {m}} and n {\displaystyle {n}} are called its dimensions . For example, the matrix A {\displaystyle {\mathbf {A} }} above

1675-566: Is divided into basins following the continents distribution : the North and South Atlantic (together approximately 75 million km / 29 million mi ), North and South Pacific (together approximately 155 million km / 59 million mi ), Indian Ocean (68 million km / 26 million mi ) and Arctic Ocean (14 million km / 5.4 million mi ). Also recognized is the Southern Ocean (20 million km / 7 million mi ). All ocean basins collectively cover 71% of

1742-532: Is known as the oceanic crust , while on land it is known as the continental crust . The former is thinner and is composed of relatively dense basalt, while the latter is less dense and mainly composed of granite. The lithosphere is composed of the crust (oceanic and continental) and the uppermost part of the mantle. The lithosphere is broken into sections called plates . Tectonic plates move very slowly (5 to 10 cm (2 to 4 inches) per year) relative to each other and interact along their boundaries. This movement

1809-472: Is not commutative , in marked contrast to (rational, real, or complex) numbers, whose product is independent of the order of the factors. An example of two matrices not commuting with each other is: whereas Besides the ordinary matrix multiplication just described, other less frequently used operations on matrices that can be considered forms of multiplication also exist, such as the Hadamard product and

1876-668: Is often denoted M ( m , n ) , {\displaystyle {\mathcal {M}}(m,n),} or M m × n ( R ) . {\displaystyle {\mathcal {M}}_{m\times n}(\mathbb {R} ).} The set of all m -by- n matrices over another field , or over a ring R , is similarly denoted M ( m , n , R ) , {\displaystyle {\mathcal {M}}(m,n,R),} or M m × n ( R ) . {\displaystyle {\mathcal {M}}_{m\times n}(R).} If m   =   n , such as in

1943-666: Is often used as a synonym for " inner product ". For example: The subtraction of two m × n matrices is defined by composing matrix addition with scalar multiplication by –1 : The transpose of an m × n matrix A is the n × m matrix A (also denoted A or A ) formed by turning rows into columns and vice versa: ( A T ) i , j = A j , i . {\displaystyle \left({\mathbf {A}}^{\rm {T}}\right)_{i,j}={\mathbf {A}}_{j,i}.} For example: Familiar properties of numbers extend to these operations on matrices: for example, addition

2010-604: Is responsible for most of the Earth's seismic and volcanic activity. Depending on how the plates interact with each other, there are three types of boundaries. The Earth's deepest trench is the Mariana Trench which extends for about 2500 km (1600 miles) across the seabed. It is near the Mariana Islands , a volcanic archipelago in the West Pacific. Its deepest point is 10994 m (nearly 7 miles) below

2077-497: Is set at the equator . The Antarctic or Southern Ocean, which reaches from 60° south to Antarctica had been omitted until 2000, but is now also recognized by the International Hydrographic Office. Nevertheless, and since ocean basins are interconnected, many oceanographers prefer to refer to one single ocean basin instead of multiple ones.   Older references (e.g., Littlehales 1930) consider

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2144-447: Is the branch of mathematics that focuses on the study of matrices. It was initially a sub-branch of linear algebra , but soon grew to include subjects related to graph theory , algebra , combinatorics and statistics . A matrix is a rectangular array of numbers (or other mathematical objects), called the entries of the matrix. Matrices are subject to standard operations such as addition and multiplication . Most commonly,

2211-418: Is the i th coordinate of f  ( e j ) , where e j = (0, ..., 0, 1, 0, ..., 0) is the unit vector with 1 in the j th position and 0 elsewhere. The matrix A is said to represent the linear map f , and A is called the transformation matrix of f . For example, the 2×2 matrix can be viewed as the transform of the unit square into a parallelogram with vertices at (0, 0) , (

2278-590: Is the inverse matrix of A . If A has no inverse, solutions—if any—can be found using its generalized inverse . Matrices and matrix multiplication reveal their essential features when related to linear transformations , also known as linear maps . A real m -by- n matrix A gives rise to a linear transformation R n → R m {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}} mapping each vector x in ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠ to

2345-413: Is the same as the set of column indices that remain. Other authors define a principal submatrix as one in which the first k rows and columns, for some number k , are the ones that remain; this type of submatrix has also been called a leading principal submatrix . Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A

2412-447: Is used in place of M . {\displaystyle {\mathcal {M}}.} Several basic operations can be applied to matrices. Some, such as transposition and submatrix do not depend on the nature of the entries. Others, such as matrix addition , scalar multiplication , matrix multiplication , and row operations involve operations on matrix entries and therefore require that matrix entries are numbers or belong to

2479-442: Is useful to consider a matrix with no rows or no columns, called an empty matrix . The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written in square brackets or parentheses , so that an m × n {\displaystyle m\times n} matrix A {\displaystyle \mathbf {A} } is represented as A = [

2546-550: The ( 1 , 3 ) {\displaystyle (1,3)} entry of the following matrix A {\displaystyle \mathbf {A} } is 5 (also denoted a 13 {\displaystyle {a_{13}}} , a 1 , 3 {\displaystyle {a_{1,3}}} , A [ 1 , 3 ] {\displaystyle \mathbf {A} [1,3]} or A 1 , 3 {\displaystyle {{\mathbf {A} }_{1,3}}} ): Sometimes,

2613-866: The Baltic Sea (with three subdivisions), the North Sea , the Greenland Sea , the Norwegian Sea , the Laptev Sea , the Gulf of Mexico , the South China Sea , and many more. The limits were set for convenience of compiling sailing directions but had no geographical or physical ground and to this day have no political significance. For instance, the line between the North and South Atlantic

2680-533: The Kronecker product . They arise in solving matrix equations such as the Sylvester equation . There are three types of row operations: These operations are used in several ways, including solving linear equations and finding matrix inverses . A submatrix of a matrix is a matrix obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct

2747-1050: The n -by- n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, for example, I 1 = [ 1 ] , I 2 = [ 1 0 0 1 ] , ⋮ I n = [ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 ] {\displaystyle {\begin{aligned}\mathbf {I} _{1}&={\begin{bmatrix}1\end{bmatrix}},\\[4pt]\mathbf {I} _{2}&={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\\[4pt]\vdots &\\[4pt]\mathbf {I} _{n}&={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}\end{aligned}}} It

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2814-421: The (matrix) product Ax , which is a vector in ⁠ R m . {\displaystyle \mathbb {R} ^{m}.} ⁠ Conversely, each linear transformation f : R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} arises from a unique m -by- n matrix A : explicitly, the ( i , j ) -entry of A

2881-804: The Atlantic and Arctic basins. The Atlantic Basin began to form around 180 million years ago, when the continent Laurasia (North America and Eurasia ) started to drift away from Africa and South America. The Pacific plate grew, and subduction led to a shrinking of its bordering plates. The Pacific plate continues to move northward. Around 130 million years ago the South Atlantic started to form, as South America and Africa started to separate. At around this time India and Madagascar rifted northwards, away from Australia and Antarctica, creating seafloor around Western Australia and East Antarctica. When Madagascar and India separated between 90 and 80 million years ago,

2948-460: The Earth's surface, and together they contain almost 97% of all water on the planet. They have an average depth of almost 4 km (about 2.5 miles). "Limits of Oceans and Seas" , published by the International Hydrographic Office in 1953, is a document that defined the ocean's basins as they are largely known today. The main ocean basins are the ones named in the previous section. These main basins are divided into smaller parts. Some examples are:

3015-617: The Mariana Islands. It is located far away from oceanic spreading centers, where oceanic crust is constantly created or destroyed. The oldest crust is estimated to be only around 200 million years old, compared to the age of Earth which is 4.6 billion years. 200 million years ago nearly all land mass was one large continent called Pangea , which started to split up. During the splitting process of Pangea, some ocean basins shrunk, such as the Pacific, while others were created, such as

3082-409: The above-mentioned associativity of matrix multiplication. The rank of a matrix A is the maximum number of linearly independent row vectors of the matrix, which is the same as the maximum number of linearly independent column vectors. Equivalently it is the dimension of the image of the linear map represented by A . The rank–nullity theorem states that the dimension of the kernel of

3149-446: The above-mentioned formula f ( i , j ) {\displaystyle f(i,j)} is valid for any i = 1 , … , m {\displaystyle i=1,\dots ,m} and any j = 1 , … , n {\displaystyle j=1,\dots ,n} . This can be specified separately or indicated using m × n {\displaystyle m\times n} as

3216-658: The best example of an inactive oceanic basin is the Gulf of Mexico, which formed in Jurassic times and has been doing nothing but collecting sediments since then. The Aleutian Basin is another example of a relatively inactive oceanic basin. The Japan Basin in the Sea of Japan which formed in the Miocene , is still tectonically active although recent changes have been relatively mild. Matrix (mathematics) In mathematics ,

3283-400: The bottom right corner of the matrix. If all entries of A below the main diagonal are zero, A is called an upper triangular matrix . Similarly, if all entries of A above the main diagonal are zero, A is called a lower triangular matrix . If all entries outside the main diagonal are zero, A is called a diagonal matrix . The identity matrix I n of size n is

3350-406: The case of square matrices , one does not repeat the dimension: M ( n , R ) , {\displaystyle {\mathcal {M}}(n,R),} or M n ( R ) . {\displaystyle {\mathcal {M}}_{n}(R).} Often, M {\displaystyle M} , or Mat {\displaystyle \operatorname {Mat} } ,

3417-503: The corresponding lower-case letters, with two subscript indices (e.g., a 11 {\displaystyle {a_{11}}} , or a 1 , 1 {\displaystyle {a_{1,1}}} ), represent the entries. In addition to using upper-case letters to symbolize matrices, many authors use a special typographical style , commonly boldface Roman (non-italic), to further distinguish matrices from other mathematical objects. An alternative notation involves

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3484-442: The entries of a matrix can be defined by a formula such as a i , j = f ( i , j ) {\displaystyle a_{i,j}=f(i,j)} . For example, each of the entries of the following matrix A {\displaystyle \mathbf {A} } is determined by the formula a i j = i − j {\displaystyle a_{ij}=i-j} . In this case,

3551-572: The individual ocean basins has fluctuated in the past due to, amongst other, tectonic plate movements. Therefore, an oceanic basin can be actively changing size and/or depth or can be relatively inactive. The elements of an active and growing oceanic basin include an elevated mid-ocean ridge , flanking abyssal hills leading down to abyssal plains and an oceanic trench . Changes in biodiversity, floodings and other climate variations are linked to sea-level, and are reconstructed with different models and observations (e.g., age of oceanic crust). Sea level

3618-474: The matrix itself is sometimes defined by that formula, within square brackets or double parentheses. For example, the matrix above is defined as A = [ i − j ] {\displaystyle {\mathbf {A} }=[i-j]} or A = ( ( i − j ) ) {\displaystyle {\mathbf {A} }=((i-j))} . If matrix size is m × n {\displaystyle m\times n} ,

3685-448: The matrix, and commonly denoted by a i , j {\displaystyle {a_{i,j}}} or a i j {\displaystyle {a_{ij}}} . Alternative notations for that entry are A [ i , j ] {\displaystyle {\mathbf {A} [i,j]}} and A i , j {\displaystyle {\mathbf {A} _{i,j}}} . For example,

3752-474: The number of rows of the right matrix. If A is an m × n matrix and B is an n × p matrix, then their matrix product AB is the m × p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B : where 1 ≤ i ≤ m and 1 ≤ j ≤ p . For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340: Matrix multiplication satisfies

3819-497: The numbering of array indexes at zero, in which case the entries of an m -by- n matrix are indexed by 0 ≤ i ≤ m − 1 {\displaystyle 0\leq i\leq m-1} and 0 ≤ j ≤ n − 1 {\displaystyle 0\leq j\leq n-1} . This article follows the more common convention in mathematical writing where enumeration starts from 1 . The set of all m -by- n real matrices

3886-406: The ocean is very slow compared to horizonal flow and observing the deep ocean is difficult. Defining the ocean basins based on connectivity of the entire ocean (depth and width) is therefore not possible. Froyland et al. (2014) defined ocean basins based on surface connectivity. This is achieved by creating a Markov Chain model of the surface ocean dynamics using short term time trajectory data from

3953-440: The oceanic basins to be the complement to the continents , with erosion dominating the latter, and the sediments so derived ending up in the ocean basins. This vision is supported by the fact that oceans lie lower than continents, so the former serve as sedimentary basins that collect sediment eroded from the continents, known as clastic sediments, as well as precipitation sediments. Ocean basins also serve as repositories for

4020-651: The rules ( AB ) C = A ( BC ) ( associativity ), and ( A + B ) C = AC + BC as well as C ( A + B ) = CA + CB (left and right distributivity ), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are m × n and n × k matrices, respectively, and m ≠ k . Even if both products are defined, they generally need not be equal, that is: A B ≠ B A . {\displaystyle {\mathbf {AB}}\neq {\mathbf {BA}}.} In other words, matrix multiplication

4087-405: The same number of rows and columns, play a major role in matrix theory. Square matrices of a given dimension form a noncommutative ring , which is one of the most common examples of a noncommutative ring. The determinant of a square matrix is a number associated with the matrix, which is fundamental for the study of a square matrix; for example, a square matrix is invertible if and only if it has

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4154-455: The skeletons of carbonate - and silica -secreting organisms such as coral reefs , diatoms , radiolarians , and foraminifera . More modern sources (e.g., Floyd 1991) regard the ocean basins more as basaltic plains, than as sedimentary depositories, since most sedimentation occurs on the continental shelves and not in the geologically defined ocean basins. The flow in the ocean is not uniform but varies with depth. Vertical circulation in

4221-673: The spreading ridges in the Indian Ocean were reorganized. The northernmost part of the Atlantic Ocean was also formed at this time when Europe and Greenland separated. About 60 million years ago a new rift and oceanic ridge formed between Greenland and Europe, separating them and initiating the formation of oceanic crust in the Norwegian Sea and the Eurasian Basin in the eastern Arctic Ocean. The area occupied by

4288-489: The surface of the ocean (plastic, biomass, water etc.) become trapped. One of these regions is for example the Atlantic garbage patch . With this approach the five main ocean basins are still the North and South Atlantic, North and South Pacific and the Arctic Ocean, but with different boundaries between the basins. These boundaries show the lines of very little surface connectivity between the different regions which means that

4355-492: The surface of the sea. The Earth's longest trench runs alongside the coast of Peru and Chile, reaching a depth of 8065 m (26460 feet) and extending for approximately 5900 km (3700 miles). It occurs where the oceanic Nazca plate slides under the continental South American plate and is associated with the upthrust and volcanic activity of the Andes. The oldest oceanic crust is in the far western equatorial Pacific, east of

4422-458: The use of a double-underline with the variable name, with or without boldface style, as in A _ _ {\displaystyle {\underline {\underline {A}}}} . The entry in the i -th row and j -th column of a matrix A is sometimes referred to as the i , j {\displaystyle {i,j}} or ( i , j ) {\displaystyle {(i,j)}} entry of

4489-467: The vertices of the unit square. The following table shows several 2×2 real matrices with the associated linear maps of ⁠ R 2 . {\displaystyle \mathbb {R} ^{2}.} ⁠ The blue original is mapped to the green grid and shapes. The origin (0, 0) is marked with a black point. Under the 1-to-1 correspondence between matrices and linear maps, matrix multiplication corresponds to composition of maps: if

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