Page 166 - A Course in Linear Algebra with Applications

150 Chapter Five: Basis and Dimension

and

2
2
3
01 = 2 + 2x - Ax 2 + x , g 2 = 1 - x + x , g 3 = 2 + 3x - x .
Let U be the subspace of P^(JR) generated by {/i, f 2} and let
W be the subspace generated by {g x, g 2, g 3}. Find bases for
the subspaces U + W and U C\W.
9. Let Ui,... ,Uk be subspaces of a vector space V. Prove
that V = U\ © • • • © Uk if and only if each element of V has a
unique expression of the form Ui + • • • + u^ where Uj belongs
to Ui.
10. Every vector space of dimension n is a direct sum of n
subspaces each of which has dimension 1. Explain why this
true.

11. If Ui,..., Uk are subspaces of a finitely generated vec-
tor space whose sum is the direct sum, find the dimension of
Ui®---®U k.

12. Let U\, U 2, U3 be subspaces of a vector space such that
Ui n U 2 = U 2nU 3 = U 2r\Ui = 0. Does it follow that
U\ + U 2 + U3 = U x © U 2 © U 3? Justify your answer.
13. Verify that all the vector space axioms hold for a quotient
space V/U.
14. Consider the linear system of Exercise 2.1.1,

x\ + 2x 2 — Sx 3 + X4 = 7
-xi + x 2 - x 3 + X4 = 4

(a) Write the general solution of the system in the form
XQ + Y, where XQ is a particular solution and Y is the general
solution of the associated homogeneous system.
(b) Identify the set of all solutions of the given linear
system as a coset of the solution space of the associated ho-
mogeneous linear system.