Question

22. (a) Find two vectors that span the null space of A 3 -1 2 -4 (b) Use the result of part (a) to find the matrix that projects vectors onto the null space of A. (c) Find two orthogonal vectors that span the null space of A. (d) Use the result of (c) to find the matrix that projects vectors onto the nul space of A. Compare this matrix with the one found in part (a). (e) Find the vector u in the null space of A for which llu -bll is a minimum, 0 with b-

Answer 1

1 1 -2 0 22(a)given A 3 2 -4 -1 2 -43 1 1 -2 0R3R,(1 1 -2 0 RREF(A)32 -4 0 -4 8 -4 R (-1/4) 1 1 -2 0 R R, (10 0 -1 0 1 -2 1 03-63R-3 0 12 1 Null space | 3-1 2-4|x=0 1 2 -4 3 100-1)| 0000i. 10 1-2 teh equation form is x1-x, = 0→ x,-x. Thus the basis for the null space ofA-

22(b)From(a) we have null space ofA- The matrix form is.A- Projection matrix is P = d(Xa)-' Ar First we need to find (A'A) -1 0 1 r_(O 2 1 0 1 -1 0 1 . 0 2 1 02-1 5 -2 1-1 0 1J110 1 AA -1 0 13 2 11 110 2 1 o 1 02 5 1 -1 0 1 1941 2 432 1 94 1 Therefore the matrix onto the null space of A- 2 4 32

(22(c) 0 0 0 v221 24-4 0 Therefore the orthogonal vectors of null space ofA -