Math 369 HW #4
Due at the beginning of class on Friday, September 26
Reading: Sections VS, S, LI, LDS.
Problems:
1. For each of the following definitions for the matrix A, decide (i) what is the rank of A and
*
*
(ii) are there zero, one, or infinitely many solutions of the equation A x = 0?


1 2 3
(a) A = 3 1 2
2 3 1


1 2 3
(b) A = 2 4 6
3 6 9


1 2
(c) A = 2 3
3 4
2. Let V be a vector space over the field F and v1 , . . . , vk ∈ V . Suppose u ∈ span(v1 , . . . , vk )
and λ ∈ F. Show that λ · u ∈ span(v1 , . . . , vk ) as well.
3. (a) Give a set of vectors that spans R2 .
(b) Give a larger set of vectors that also spans R2 .
(c) As you might guess from part (b), we can make arbitrarily large spanning sets for R2 .
How small can a spanning set for R2 be? In other words, what is the fewest number of
vectors there can be in a spanning set for R2 ?
4. As a reminder, here is some notation for some standard vector spaces:
• For a closed interval [a, b], the set C 0 ([a, b]) is the set of continuous functions on [a, b].
Similarly, C 0 ((a, b)) is the set of continuous functions on the open interval (a, b), etc.
• For an interval [a, b], the set C ∞ ([a, b]) is the set of infinitely differentiable functions on
[a, b], and likewise for open intervals. In particular, C ∞ ((−∞, ∞)) is the set of infinitely
differentiable functions on (−∞, ∞) = R.
• P(R)[z] is the set of polynomials in the variable z with coefficients in R and Pd (R)[z] is
the set of polynomials of degree at most d.
Now, in each of the following, S describes a subset of a vector space V (you don’t have to
prove that V really is a vector space). For each, determine whether S is a subspace of V and
justify your answer.
 

 0

(a) V = R3 , S = y  : y, z ∈ R


z
1
 

 1

(b) V = R3 , S = y  : y, z ∈ R


z
(c) V = C 0 ([1, 5]), S = {f ∈ V : f (3) = 2}
(d) V = C 0 ([1, 5]), S = {f ∈ V : f (3) = 0}
(e) V = C ∞ ((−∞, ∞)), S = {f ∈ V : f 0 (x) − f 00 (x) = 0}
(f) V = C 0 ([−2, 2]), S = {f ∈ V : f (−x) = f (x) for all x ∈ [−2, 2]}
5. (Optional) Let V be a vector space, and let U and W be subspaces of V .
(a) Prove that the intersection U ∩ W = {v ∈ V : v ∈ U and v ∈ W } is also a subspace of
V.
(b) Give an example showing that the union U ∪ W = {v ∈ V : v ∈ U or v ∈ W } is not
necessarily a subspace.
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