Linear Algebra, 2020a

Section III. Basis and Dimension 151
4.31 Prove that if V = W 1 ⊕ ...⊕ W k then W i ∩ W j is trivial whenever i ≠ j. This
shows that the first half of the proof of Lemma 4.15 extends to the case of more
than two subspaces. (Example 4.19 shows that this implication does not reverse;
the other half does not extend.)
4.32 Recall that no linearly independent set contains the zero vector. Can an
independent set of subspaces contain the trivial subspace?
̌ 4.33 Does every subspace have a complement?
̌ 4.34 Let W 1 ,W 2 be subspaces of a vector space.
(a) Assume that the set S 1 spans W 1 , and that the set S 2 spans W 2 . Can S 1 ∪ S 2
span W 1 + W 2 ? Must it?
(b) Assume that S 1 is a linearly independent subset of W 1 and that S 2 is a linearly
independent subset of W 2 . Can S 1 ∪ S 2 be a linearly independent subset of
W 1 + W 2 ? Must it?
4.35 When we decompose a vector space as a direct sum, the dimensions of the
subspaces add to the dimension of the space. The situation with a space that is
given as the sum of its subspaces is not as simple. This exercise considers the
two-subspace special case.
(a) For these subspaces of M 2×2 find W 1 ∩ W 2 , dim(W 1 ∩ W 2 ), W 1 + W 2 , and
dim(W 1 + W 2 ).
( )
( )
0 0
0 b
W 1 = { | c, d ∈ R} W 2 = { | b, c ∈ R}
c d
c 0
(b) Suppose that U and W are subspaces of a vector space. Suppose that the
sequence 〈⃗β 1 ,...,⃗β k 〉 is a basis for U ∩ W. Finally, suppose that the prior
sequence has been expanded to give a sequence 〈⃗μ 1 ,...,⃗μ j , ⃗β 1 ,...,⃗β k 〉 that is a
basis for U, and a sequence 〈⃗β 1 ,...,⃗β k , ⃗ω 1 ,..., ⃗ω p 〉 that is a basis for W. Prove
that this sequence
〈⃗μ 1 ,...,⃗μ j , ⃗β 1 ,...,⃗β k , ⃗ω 1 ,..., ⃗ω p 〉
is a basis for the sum U + W.
(c) Conclude that dim(U + W) =dim(U)+dim(W)−dim(U ∩ W).
(d) Let W 1 and W 2 be eight-dimensional subspaces of a ten-dimensional space.
List all values possible for dim(W 1 ∩ W 2 ).
4.36 Let V = W 1 ⊕···⊕W k and for each index i suppose that S i is a linearly
independent subset of W i . Prove that the union of the S i ’s is linearly independent.
4.37 A matrix is symmetric if for each pair of indices i and j, the i, j entry equals
the j, i entry. A matrix is antisymmetric if each i, j entry is the negative of the j, i
entry.
(a) Give a symmetric 2×2 matrix and an antisymmetric 2×2 matrix. (Remark.
For the second one, be careful about the entries on the diagonal.)
(b) What is the relationship between a square symmetric matrix and its transpose?
Between a square antisymmetric matrix and its transpose?
(c) Show that M n×n is the direct sum of the space of symmetric matrices and the
space of antisymmetric matrices.
4.38 Let W 1 ,W 2 ,W 3 be subspaces of a vector space. Prove that (W 1 ∩ W 2 )+(W 1 ∩
W 3 ) ⊆ W 1 ∩ (W 2 + W 3 ). Does the inclusion reverse?