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9 Jan 2018

1 3 5 -47 | 20 4 41 5. Let A= | 1-2 0 -4 -4 | 0 2 2 - 4 (a) Find an orthogonal basis for the column space of A. (b) Find a basis for the orthogonal complement of the column space of A.

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Patrina Schowalter
Patrina SchowalterLv2
11 Jan 2018
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4. (12 points) Let the matrix A and its reduced row echelon form R be given by the following. A= [ 1 -2 2 -3 1 -2 4 -1 4 1 -3] [1001 1] -4 0 0 1 0 1 1 7 -1 0 0 1 1 -1 | = R. (a) Find a basis for the column space of A. (b) Find a basis for the row space of A. (c) Find a basis for the null space of A. (d) Verify that the dimension of the column space of A plus the dimension of the null space of A is equal to the number of columns of A.

5. Consider the vector space P2(R) together with inner product =P(-1),(-1) + p(0,0) + p(1)g(1). (5) (a) Use the Gram-Schmidt procedure to find an orthogonal basis for W = Span{1, x + 32²}. [5] (b) Find the orthogonal projection of 2 + 4x onto W.

(10] 1. Given that /1 0 2 1 -1 2 0 3 0 1 A4 07 2-1 0 0 1 3 1 / has reduced row echelon form R= (1 000 17 0 0 1 0 -11 0001 4: OOOOO find a basis for each of the following: (a) The row space of A. (b) The column space of A. (c) The null space of A.

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