Homework 6
Due: Wednesday, October 15
Before we start, here is some notation for certain fundamental vector spaces:
• C[0, 1] is the set of continuous, R-valued functions on the unit interval. C[a, b] is defined
analogously.
• C ∞ [−∞, ∞] is the set of infinitely differentiable functions from R to R.
• P (R)[ z] is the set of polynomials with real coefficients in a variable z.
• Pd (R)[ z] is the set of polynomials with real coefficients in a variable z of degree at most d.
 
x
To save space, we will sometimes write, e.g., ( x y z) T for  y. (The T means transpose.)
z
1. Let V be a vector space, and let U and W be subspaces.
(a) Prove that the intersection U ∩ W is also a subspace of V.
(b) Give an example showing that U ∪ W need not be a subspace of V. (H INT: Think about
two lines in R3 .)
2. Give an example of a vector space V and a subset U such that U is closed under scalar
multiplication, but U is not a subspace of V.
3. Each of the following describes a subset S of a vector space V. (You may assume that V
really is a vector space.) In each case, is S a subspace of V? Justify your answer.
(a) V = R3 , S = {(0, a, b) T : a, b ∈ R}.
(b) V = R3 , S = {(1, a, b) T : a, b ∈ R}.
(c) V = C[0, 10], S = { f ∈ V : f (4) = 3}.
(d) V = C[0, 10], S = { f ∈ V : f (4) = 0}.
(e) V = C ∞ [−∞, ∞], S = { f ∈ V : f ′ ( x) − f ′′ ( x) = 0}. (H INT: Don’t compute the set S
explicitly!)
(f) V = C[−1, 1], S = { f ∈ V : f (− x) = f ( x)}.
4. Which of the following sets are spanning sets for R3 ? In each case, if the set spans R3 , prove
it; if not, write down a vector which is not in the span of the set in question.
(a) {(1, 1, 1) T }
(b) {(1, 2, 0) T , (0, 1, 0) T , (0, 2, 1) T }
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008
(c) {(1, 2, 0) T , (0, 1, 0) T , (0, 2, 1) T , (1, 2, 3) T }
(d) {(1, 2, −2) T , (2, 3, −2) T , (2, 2, 0) T , (3, 4, −2) T }
(e) {(1, 2, −2) T , (2, 3, −2) T , (2, 2, 0) T }
(H INT: Sometimes you are better off thinking before computing.)
5. Let V be a vector space over a field F. Suppose v1 , v2 ∈ V, and λ ∈ F is nonzero. Show that
span (v1 , v2 ) = span(λv1 , v2 ).
Professor Dan Bates
Colorado State University
M369 Linear Algebra
Fall 2008