MATH 121A University of California Linear Equations Algebra Problems

please solve these 6 questions

please writing clearly or typing

thank u very much

Math 121A
Homework 3
Explain all solutions/justify your work. All vector spaces are over a field F which may be either
R or C unless stated otherwise.
1. Assume that V is a one-dimensional vector space. Prove that the only subspaces of V are {0}
and V .
2. Assume that U and W are one-dimensional subspaces of a vector space V . Prove that U 6=
W ⇔ U ∩ W = {0}.
3. Let U = {p ∈ P3 (R)|p(1) = p(2)}. (U is a subspace – you do not need to prove this.)
(a) Find a basis for U .
(b) Extend the basis of U in part (a) to a basis of P3 (R).
4. Assume that U and V are 4-dimensional subspaces of R6 , and assume that U + V 6= R6 . Find
all possible dimensions of U ∩ V .
5. Assume that T : V → W is a linear map of finite-dimensional vector spaces. Let (v1 , . . . , vn )
be a list of vectors in V . Assume that (T v1 , . . . , T vn ) is linearly independent. Prove that
(v1 , . . . , vn ) is linearly independent.
6. Define the function T : R3 → P2 (R) by T (a, b, c) = a + b(1 − x2 ) + cx2 . Prove that T is a linear
map.