Linear Algebra

Section III. Basis and Dimension 139
(b) Suppose that U and W are subspaces of a vector space. Suppose that the
sequence 〈 � β1,..., � βk〉 is a basis for U ∩ W . Finally, suppose that the prior
sequence has been expanded to give a sequence 〈�µ1,...,�µj, � β1,..., � βk〉 that is a
basis for U, and a sequence 〈 � β1,..., � βk,�ω1,...,�ωp〉 that is a basis for W .Prove
that this sequence
〈�µ1,...,�µj, � β1,..., � βk,�ω1,...,�ωp〉
is a basis for for the sum U + W .
(c) Conclude that dim(U + W )=dim(U)+dim(W ) − dim(U ∩ W ).
(d) Let W1 and W2 be eight-dimensional subspaces of a ten-dimensional space.
List all values possible for dim(W1 ∩ W2).
4.36 Let V = W1 ⊕ ...⊕ Wk and for each index i suppose that Si is a linearly
independent subset of Wi. Prove that the union of the Si’s is linearly independent.
4.37 Amatrixissymmetric if for each pair of indices i and j, thei, 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 diagional.)
(b) What is the relationship between a square symmetric matrix and its transpose?
Between a square antisymmetric matrix and its transpose?
(c) Show that Mn×n is the direct sum of the space of symmetric matrices and
the space of antisymmetric matrices.
4.38 Let W1,W2,W3 be subspaces of a vector space. Prove that (W1 ∩W2)+(W1∩ W3) ⊆ W1 ∩ (W2 + W3). Does the inclusion reverse?
4.39 The example of the x-axis and the y-axis in R 2 shows that W1 ⊕ W2 = V does
not imply that W1 ∪ W2 = V .CanW1⊕W2 = V and W1 ∪ W2 = V happen?
� 4.40 Our model for complementary subspaces, the x-axis and the y-axis in R 2 ,
has one property not used here. Where U is a subspace of R n we define the
orthocomplement of U to be
U ⊥ = {�v ∈ R n � � �v �u =0forall�u∈ U}
(read “U perp”).
(a) Find the orthocomplement of the x-axis in R 2 .
(b) Find the orthocomplement of the x-axis in R 3 .
(c) Find the orthocomplement of the xy-plane in R 3 .
(d) Show that the orthocomplement of a subspace is a subspace.
(e) Show that if W is the orthocomplement of U then U is the orthocomplement
of W .
(f) Prove that a subspace and its orthocomplement have a trivial intersection.
(g) Conclude that for any n and subspace U ⊆ R n we have that R n = U ⊕ U ⊥ .
(h) Show that dim(U)+dim(U ⊥ ) equals the dimension of the enclosing space.
� 4.41 Consider Corollary 4.13. Does it work both ways — that is, supposing that
V = W1 + ···+ Wk, isV = W1 ⊕ ...⊕ Wk if and only if dim(V )=dim(W1) +
···+dim(Wk)?
4.42 We know that if V = W1 ⊕ W2 then there is a basis for V that splits into a
basis for W1 and a basis for W2. Can we make the stronger statement that every
basis for V splits into a basis for W1 and a basis for W2?
4.43 We can ask about the algebra of the ‘+’ operation.
(a) Is it commutative; is W1 + W2 = W2 + W1?
(b) Is it associative; is (W1 + W2)+W3 = W1 +(W2 + W3)?