Math 205a Quiz 6 Worksheet - Bates College - 2007
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1Math 205A
Quiz 6, page 1
November 2, 2007
NAME
1. Suppose b has components b
, b
, b
, A is a 3x4 matrix and the augmented matrix corresponding to
1
2
3
1 4 6 8
b
2b
1
3
.
the equation Ax = b is row equivalent to
0 0 1 1
b
+ 4b
2
3
0 0 0 k 3b
+ 2b
+ b
1
2
3
1A. Suppose k = 1. What conditions (if any) must b
, b
and b
satisfy in order for b to be in Col(A)?
1
2
3
Explain!
3
1B. So, if k = 1, is Col(A) all of R
? Explain!
1C. Suppose k = 1. Find vectors that span the nullspace of A. Hint: Think about the way we write
the solutions of the homogeneous equation Ax = 0 in “parametric form”.
1D. Suppose k = 0. What conditions (if any) must b
satisfy in order for b to be in Col(A)?
, b
and b
1
2
3
Explain!
3
1E. So, if k = 0, is Col(A) all of R
? Explain!
2. Let F be the vector space of all continuous functions f : R → R, as discussed in class.
2A. Is the set H = {f ∈ F | the graph of f passes through the point (0, 3)} closed under vector addi-
tion? Prove it or give a counterexample.
2B. Is the set G = {f ∈ F | the graph of f passes through the point (3, 0)} closed under vector addi-
tion? Prove it or give a counterexample.
2C. Which (if either) of H or G is a subspace of F?
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