323 Math Quiz 6 March 12, 2009
Name: KEY Show your work for full credit.
1. Let W we the set of all vectors of the form
2a - 3b a + 5b - c 0 -2b + c
where a, b and c represent arbitrary real numbers. Find a set S of vectors that spans W .
2a - 3b a + 5b - c 0 -2b + c
= a
2 1 0 0
+ b
-3 5 0 -2
+ c
0 -1 0 1
-3 0 2 1 5 -1 S = , , 0 0 0
0
-2
1
2. Let A =
1 -5 3 0 . 0 0 -1 0
(a) Find a set of vectors that span ColA. ColA = Span (b) Find a set of vectors span that NulA. 1 -5 3 0 0 0 -1 0 x1 x2 x3 x4
1 0
,
-1 3
-
1 -5 0 0 0 0 1 0
= is = is
5x2 free 0 free
x=
5x2 x2 0 x4
= x2
5 1 0 0
+ x4
0 0 0 1
0 5 1 0 NulA = Span , 0 0
0
1
3. Prove that the kernel of a linear transformation T from a vector space V ...