Page 34 - Linear Algebra Done Right

Chapter 1. Vector Spaces
20
12.
additive identity? Which subspaces have additive inverses?
Prove or give a counterexample: if U 1 ,U 2 ,W are subspaces of V
13. Does the operation of addition on the subspaces of V have an
such that
U 1 + W = U 2 + W,
then U 1 = U 2 .
14. Suppose U is the subspace of P(F) consisting of all polynomials
p of the form
5
2
p(z) = az + bz ,
where a, b ∈ F. Find a subspace W of P(F) such that P(F) =
U ⊕ W.

15. Prove or give a counterexample: if U 1 ,U 2 ,W are subspaces of V
such that
V = U 1 ⊕ W and V = U 2 ⊕ W,

then U 1 = U 2 .