4305 Math bg & bu: Topics in linear algebra Summer 2005
Sampe test 1 (alpha)
Wed., 06/08/05 . Show all your work. You have 70 mins. to work on this test. Use the blanksheets provided to write your answers. Please do not write your answers on this sheet. Turn in this sheet with your answers.
Name:
1. Find a matrix A whose row space contains the row vector y t = [2 1 1] and whose null space contains the (column) vector x = [1 2 0] t , or prove that no such matrix can exist.
2. Find bases for the four fundamental subspaces associated to the following matrix: 2 1 1 0 2 1 . A= 4 2 2 0 6 3
3. Use the method of leastsquares to fit a line y = mx + b through the following points (x, y ): (2, 3) , (0, 1) , (2, 0) , (1, 1) .
4. Let
1 1 1 A = 0 2 0
.
Find projection the matrix onto N (A t ).
There is one more problem on the next page.
1
5. Let
0 2 0 0 3 0 A= 1 2 0 3 5 1
4 6 0 1
.
(a) Find a permutation matrix P1 such that 1 2 0 0 3 0 P1 A = 0 2 0 3 5 1 (b) Find an elementary matrix E1 such that
0 6 4 1
.
1 0 E1 P1 A = 0 0
2 0 3 0 2 0 1 1
0 6 4 1
.
(c) Find a permutation matrix P2 such that
1 0 P2 E1 P1 A = 0 0
2 00 1 1 1 2 0 4 3 06
.
(d) Find an elementary matrix E2 suc...