Ch5_matrices

Ch5_matrices - Chapter 5 Matrices Matlab began as a matrix...

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Chapter 5 Matrices Matlab began as a matrix calculator. The Cartesian coordinate system was developed in the 17th century by the French mathematician and philosopher Ren´ e Descartes. A pair of numbers corre- sponds to a point in the plane. We will display the coordinates in a vector of length two. In order to work properly with matrix multiplication, we want to think of the vector as a column vector, So x = ± x 1 x 2 denotes the point x whose first coordinate is x 1 and second coordinate is x 2 . When it is inconvenient to write a vector in this vertical form, we can anticipate Matlab notation and use a semicolon to separate the two components, x = ( x 1 ; x 2 ) For example, the point labeled x in figure 5.1 has Cartesian coordinates x = ( 2; 4 ) Arithmetic operations on the vectors are defined in natural ways. Addition is defined by x + y = ± x 1 x 2 + ± y 1 y 2 = ± x 1 + y 1 x 2 + y 2 Multiplication by a single number, or scalar , is defined by sx = ± sx 1 sx 2 Copyright c ± 2009 Cleve Moler Matlab R ± is a registered trademark of The MathWorks, Inc. TM August 8, 2009 1
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2 Chapter 5. Matrices A 2-by-2 matrix is an array of four numbers arranged in two rows and two columns. A = ± a 1 , 1 a 1 , 2 a 2 , 1 a 2 , 2 or A = ( a 1 , 1 a 1 , 2 ; a 2 , 1 a 2 , 2 ) For example A = ± 4 - 3 - 2 1 -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 x Ax Figure 5.1. Matrix multiplication transforms lines through x to lines through Ax. Matrix-vector multiplication by a 2-by-2 matrix A transforms a vector x to a vector Ax , according to the definition Ax = ± a 1 , 1 x 1 + a 1 , 2 x 2 a 2 , 1 x 1 + a 2 , 2 x 2 For example ± 4 - 3 - 2 1 ¶± 2 4 = ± 4 · 2 - 3 · 4 - 2 · 2 + 1 · 4 = ± - 4 0 The point labeled x in figure 5.1 is transformed to the point labeled Ax . Matrix- vector multiplications produce linear transformations. This means that for scalars s and t and vectors x and y , A ( sx + ty ) = sAx + tAy
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3 This implies that points near x are transformed to points near Ax and that straight lines in the plane through x are transformed to straight lines through Ax . Our definition of matrix-vector multiplication is the usual one involving the dot product of the rows of A , denoted a i, : , with the vector x . Ax = ± a 1 , : · x a 2 , : · x An alternate, and sometimes more revealing, definition uses linear combinations of the columns of A , denoted by a : ,j . Ax = x 1 a : , 1 + x 2 a : , 2 For example ± 4 - 3 - 2 1 ¶± 2 4 = 2 ± 4 - 2 + 4 ± - 3 1 = ± - 4 0 The transpose of a column vector is a row vector, denoted by x T . The trans- pose of a matrix interchanges its rows and columns. For example, x T = ( 2 4 ) A T = ± 4 - 2 - 3 1 Vector-matrix multiplication can be defined by x T A = A T x That is pretty cryptic, so if you have never seen it before, you might have to ponder it a bit. Matrix-matrix multiplication,
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Ch5_matrices - Chapter 5 Matrices Matlab began as a matrix...

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