Math 115A
Homework 2
Due Tuesday, July 11, 2023
I Subspaces, and the span of a set of vectors
1. Let F be a field, and recall that F n, the set of all n-tuples of elements of F , is a vector
space over F . Let
W1 = { (a1, a2, . . . , an) œ F n | a1 + a2 + · · ·+ an = 0 }
and let
W2 = { (a1, a2, . . . , an) œ F n | a1 + a2 + · · ·+ an = 1 }
Prove that W1 is a subspace of F n, but W2 is not a subspace of F n.
2. Let F be a field, and recall that P (F ) denotes the vector space of all polynomials with
coecients in F . Let n be a positive integer. Is the set
W = { f œ P (F ) | f = 0 or deg(f) = n }
a subspace of P (F )? Prove your answer.
3. Let X be a nonempty set and let F be a field, and recall that F(X,F ) denotes the
vector space of all functions f :X æ F . Fix some x0 œ X. Let
W = { f œ F(X,F ) | f(x0) = 0 } .
Prove that W is a subspace of F(X,F ).
4. Let F and K be fields, and recall that F(K,F ) denotes the vector space of all functions
from K to F . A function g:K æ F is called an even function if
g(≠x) = g(x) for all x œ K,
and is called an odd function if
g(≠x) = ≠g(x) for all x œ K.
(These are the same kind of even and odd functions you learned about way back in
high school algebra: for functions f :Ræ R, f is even i its graph is symmetric across
the y-axis, and f is odd i its graph is symmetric about the origin.) Let W1 be the set
of all odd functions in F(K,F ), and W2 be the set of all even functions in F(K,F ).
Prove that both W1 and W2 are subspaces of F(K,F ).
5. Let V be a vector space over a field F , and let S1 and S2 be subsets of V such
that S1 ™ S2. Prove that span(S1) ™ span(S2). Deduce that if span(S1) = V then
span(S2) = V .
II Linear dependence, linear independence, and bases
6. Label each of the following statements as true or false. If true, explain why briefly (no
need to give a rigorous proof). If false, either give a counterexample or say how the
statement should be modified to make it true.
(a) If S is a linearly dependent set, then each vector in S is a linear combination of
other vectors in S.
(b) Subsets of linearly dependent sets are linearly dependent.
(c) Subsets of linearly independent sets are linearly independent.
(d) If a1x1 + · · ·+ anxn = 0 and x1, . . . , xn are linearly independent, then a1 = · · · =
an = 0.
(e) The zero vector space {0} does not have a basis.
7. Is this a linearly independent subset of R3?
{(1, 4,≠6), (1, 5, 8), (2, 1, 1), (0, 1, 0)}
Justify your answer.
8. The vectors u1 = (2,≠3, 1), u2 = (1, 4,≠2), u3 = (≠8, 12,≠4), u4 = (1, 37,≠17), and
u5 = (≠3,≠5, 8) span R3. (You don’t need to prove this.) Find a subset of the set
G = {u1, u2, u3, u4, u5} that is a basis of R3.
9. Let F be a field. Find bases of the following two subspaces of F 5:
W1 = { (a1, a2, a3, a4, a5) œ F 5 | a1 ≠ a3 ≠ a4 = 0 } and
W2 = { (a1, a2, a3, a4, a5) œ F 5 | a2 = a3 = a4 and a1 + a5 = 0 }
What are the dimensions of W1 and W2?
10. Let V and W be vector spaces over a field F , of dimensions m and n, respectively.
Define a vector space Z over F as in Homework 1:
Z = { (v, w) | v œ V and w œ W }
with addition and scalar multiplication defined component-wise. Determine the di-
mension of Z. Prove your result.
11. Fix some positive integer n, and some a œ R. The set
W = { f œ Pn(R) | f(a) = 0 }
is a subspace of Pn(R), the space of polynomials with real coecients of degree Æ n.
(You do not need to prove this.) Determine the dimension of W .