For any matrix A, let N(A) denote its null space. In the real space Rn, consider

Question

For any matrix A, let N(A) denote its null space. In the real space Rn, consider the inner product (x, y) = x1y1 + + Xnyn and 2-norm |x| = (x,x),1/2 for every vectors x = [x1 xn] T and y = [y1 yn]T in Rn. Suppose S is a subspace of Rn. Let S1 be the orthogonal complement of S in SR". For the following matrix A (a) find a basis beta for N(A), and (b) check if x = [0 0 1 0 0 ] T is a vector with the smallest 2-norm satisfying Ax - [1 2 2 1] T and explain why. (15%) A = [3 3 1 3 3 2 4 2 4 2 0 3 2 3 0 -1 1 1 1 -1]

Explanation / Answer

N(A) has the orthogonal complement as the row space of A, sothis is easy. Just calculate a basis for the row space. To do that,convert the matrix into row echelon form and select the non zerorows.

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