# Schaum's Outline of Theory and Problems of Linear Algebra by Seymour Lipschutz

By Seymour Lipschutz

This ebook doesn't fairly educate Linear Algebra. it's totally superficial and never attention-grabbing. test Jim Hefferon's publication in its place and you'll have a transparent textual content that teaches Linear Algebra for real.

Anyway, there are numerous workouts within the ebook -- that's one cause to shop for it, I think...

Read or Download Schaum's Outline of Theory and Problems of Linear Algebra (Schaum's Outlines) PDF

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Extra info for Schaum's Outline of Theory and Problems of Linear Algebra (Schaum's Outlines)

Example text

5. Let r Â s denote an r Â s matrix. Find the sizes of those matrix products that are de®ned: (a) (b) 2 Â 33 Â 4; 4 Â 11 Â 2, (c) 1 Â 23 Â 1; (d) 5 Â 22 Â 3, (e) 4 Â 43 Â 3 ( f ) 2 Â 22 Â 4 In each case, the product is de®ned if the inner numbers are equal, and then the product will have the size of the outer numbers in the given order. 6. 1 Let A  2 (a) (c) not de®ned, (d) 5 Â 3, 3 À1   2 and B  3 (e) not de®ned (f) 2 Â 4 0 À2  À4 . Find: (a) AB, (b) BA. 6 Since A is a 2 Â 2 and B a 2 Â 3 matrix, the product AB is de®ned and is a 2Â 3 matrix.

2 0 À5 À2 Since A is a 2 Â 3 and B a 3 Â 4 matrix, the product AB is de®ned and is a 2 Â 4 matrix. 8. Find: (a) (a) 1 À3 6 5   2 , À7  (b) 2 À7   6 , 5 1 À3  (c) 2; À7  1 6 . À3 5 The ®rst factor is 2 Â 2 and the second is 2 Â 1, so the product is de®ned as a 2 Â 1 matrix:        1 6 2 2 À 42 À40   À3 5 À7 À6 À 35 À41 (b) The product is not de®ned, since the ®rst factor is 2 Â 1 and the second factor is 2 Â 2. 9. Clearly 0A  0 and A0  0, where the 0's are zero matrices (with possibly different sizes).

11. 2(ii): AB  C  AB  AC. Let A  aij , B  bjk , C  cjk , and let D  B  C  djk , E  AB  eik , F  AC   fik . Then djk  bjk  cjk ; eik  m P j1 aij bjk ; fik  m P j1 aij cjk Lipschutz−Lipson:Schaum’s Outline of Theory and Problems of Linear Algebra, 3/e 2. Algebra of Matrices 46 © The McGraw−Hill Companies, 2004 Text ALGEBRA OF MATRICES [CHAP. 2 Thus the ik-entry of the matrix AB  AC is eik  fik  m P j1 aij bjk  m P j1 aij cjk  m P j1 aij bjk  cjk  On the other hand, the ij-entry of the matrix AD  AB  C is ai1 d1k  ai2 d2k  Á Á Á  aim dmk  m P j1 aij djk  m P j1 aij bjk  cjk  Thus AB  C  AB  AC, since the corresponding elements are equal.