SSM-Ch07 - Chap. 7 Sec. 7.1 Linear Algebra: Matrices,...

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Chap . 7 Linear Algebra : Matrices , Vectors , Determinants , Linear Systems Sec . 7 . 1 Matrices , Vectors : Addition and Scalar Multiplication Problem Set 7 . 1 . Page 277 5 . Matrix addition , scalar multiplication . Calculate 3 C u 8 D as follows. First multiply C by 3, then D by 8. This gives 3 C u 3 0 2 2 4 1 3 u 0 6 6 12 3 9 and 8 D u 8 6 1 u 4 7 u 8 3 u 48 8 u 32 56 u 64 24 . The resulting matrices have the same size as the given ones, namely 3 U 2 (3 rows, 2 columns) because scalar multiplication does not alter the size of a matrix. Hence the operations of addition and subtraction are defined for these matrices, and you obtain the result by subtracting the entries of 8 D from the corresponding ones of 3 C , that is, 3 C u 8 D u 0 u 48 6 u 8 6 ± 32 12 u 56 3 ± 64 9 u 24 u u 48 u 2 38 u 44 67 u 15 . Matrix addition and scalar multiplication have properties quite similar to those of the addition and multiplication of numbers – note well that we are presently concerned with multiplying the entries of a
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Chap. 7 Linear Algebra: Matrices, Vectors, Determinants, Linear Systems 67 matrix by a number (the same number), and we are not yet multiplying a matrix by another matrix; this will be done in the next section. The next task in the problem illustrates formula (4c) on p. 276, showing that you get the same result by multiplying a matrix by 3 and the result then by 4, or by multiplying the matrix at once by 3 u 4 u 12 : 4 u 3 A U u u 4 u 3 U A u 12 A u 12 3 0 4 U 1 2 2 6 5 U 4 u 36 0 48 U 12 24 24 72 60 U 48 Finally, B U 1 10 A is also defined since both A and B are square matrices, of size 3 U 3, and so is 1 10 A . You obtain B U 1 10 A u 0 U 5 U 3 U 5 2 4 U 3 4 0 U 0.3 0.0 0.4 U 0.1 0.2 0.2 0.6 0.5 U 0.4 u U 0.3 U 5.0 U 3.4 U 4.9 1.8 3.8 U 3.6 3.5 0.4 . 7 . Vectors are special matrices, having a single row or a single column, and operations with them are the same as for general matrices (and somewhat simpler). u and v are column vectors, and they have the same number of components, they are of the same size 3 U 1. Hence they can be added. You obtain in this problem 33 u u 33 2 0 U 1 u 66 0 U 33 . For the next two tasks you obtain the same result because 4 v ± 9 4 u u 4 v ± 9 u u 4 U 4.5 0.8 1.2 ± 9 2 0 U 1 u U 18 3.2 4.8 ± 18 0 U 9 u 0 3.2 U 4.2 . Finally, u U v u 2 0 U 1 U U 4.5 0.8 1.2 u 6.5 U 0.8 U 2.2 . 15 . Proof of ( 3a ). A and B are assumed to be general 3 U 2 matrices. Hence you have A u a 11 a 12 a 21 a 22 a 31 a 32 and B u b 11 b 12 b 21 b 22 b 31 b 32 . Now use the definition of matrix addition to obtain A ± B u a 11 ± b 11 a 12 ± b 12 a 21 ± b 21 a 22 ± b 22 a 31 ± b 31 a 32 ± b 32 and
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68 Linear Algebra, Vector Calculus Part B B u A U b 11 u a 11 b 12 u a 12 b 21 u a 21 b 22 u a 22 b 31 u a 31 b 32 u a 32 . Now remember what you want to prove. You want to prove that
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This note was uploaded on 04/08/2012 for the course MT 423 taught by Professor M.lee during the Spring '12 term at National Taiwan University.

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SSM-Ch07 - Chap. 7 Sec. 7.1 Linear Algebra: Matrices,...

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