Page 50 - Linear Algebra Done Right

Chapter 2. Finite-Dimensional Vector Spaces
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11.
such that dim U = dim V. Prove that U = V.
Suppose that p 0 ,p 1 ,...,p m are polynomials in P m (F) such that
12. Suppose that V is finite dimensional and U is a subspace of V
p j (2) = 0 for each j. Prove that (p 0 ,p 1 ,...,p m ) is not linearly
independent in P m (F).
8
13. Suppose U and W are subspaces of R such that dim U = 3,
8
dim W = 5, and U + W = R . Prove that U ∩ W ={0}.
9
14. Suppose that U and W are both five-dimensional subspaces of R .
Prove that U ∩ W ={0}.
15. You might guess, by analogy with the formula for the number
of elements in the union of three subsets of a finite set, that
if U 1 ,U 2 ,U 3 are subspaces of a finite-dimensional vector space,
then
dim(U 1 + U 2 + U 3 )

= dim U 1 + dim U 2 + dim U 3
− dim(U 1 ∩ U 2 ) − dim(U 1 ∩ U 3 ) − dim(U 2 ∩ U 3 )
+ dim(U 1 ∩ U 2 ∩ U 3 ).

Prove this or give a counterexample.

16. Prove that if V is finite dimensional and U 1 ,...,U m are subspaces
of V, then
dim(U 1 + ··· + U m ) ≤ dim U 1 +· · ·+ dim U m .

17. Suppose V is finite dimensional. Prove that if U 1 ,...,U m are
subspaces of V such that V = U 1 ⊕ ··· ⊕ U m , then
dim V = dim U 1 +· · ·+ dim U m .

This exercise deepens the analogy between direct sums of sub-
spaces and disjoint unions of subsets. Specifically, compare this
exercise to the following obvious statement: if a finite set is writ-
ten as a disjoint union of subsets, then the number of elements in
the set equals the sum of the number of elements in the disjoint
subsets.