By Hugo. Rossi

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**Example text**

50 Linear Functions /. We want to find all vectors b of the form x\ and we (1-27) b = want to find system corresponding Now basis for such vectors. 27) is the to the matrix A whose columns are the vectors Trie sPan f these vectors is Just trie range of A. If of elementary matrices row reducing A, then any vector product Thus by b is in the range of A if and only if Pb is in the range of PA. row reduction we should easily be able to solve our problem. i, v2 P is a A v4 . = The end result of row '11 1 PA=| ' 0 l ~!

The b's an1x"=ZpJ1bJ a2x"=2pj2bJ = . . Pb , 2 p/bJ b" are known. ,x" are . . , = EXERCISES 13. Compute the products AB: 3^ (b) A= j 3; 38 Linear Functions 1. 4\ 2 (c) A B (6, 6, 3, 2,1,) = -1 8 \ V (d) A 4 2 8 6 1 2 -3 0 0 1 1 0 1 0 -1 -1 B = 14. Compute the products BA for the matrices A, B of Exercise 13. 1 5. Compute the matrix corresponding to the sequence of row operations which row reduce the matrices of Exercise 5. given m x n matrix A find conditions equation Ax b has a solution. as given in Exercise 13(a).

We want to find all vectors b of the form x\ and we (1-27) b = want to find system corresponding Now basis for such vectors. 27) is the to the matrix A whose columns are the vectors Trie sPan f these vectors is Just trie range of A. If of elementary matrices row reducing A, then any vector product Thus by b is in the range of A if and only if Pb is in the range of PA. row reduction we should easily be able to solve our problem. i, v2 P is a A v4 . = The end result of row '11 1 PA=| ' 0 l ~! b1 = + b2 - f b3 dim V < 3.