Linear Algebra

118 Chapter 2. Vector Spaces
� 1.21 Find a basis for each.
(a) The subspace {a2x 2 �
+ a1x + a0 � a2 − 2a1 = a0} of P2
(b) The space of three-wide row vectors whose first and second components add
to zero
(c) This subspace of the 2×2 matrices
� �
a b ��
{ c − 2b =0}
0 c
1.22 Check Example 1.6.
� 1.23 Find the span of each set and then find a basis for that span.
(a) {1+x, 1+2x} in P2 (b) {2 − 2x, 3+4x 2 } in P2
� 1.24 Find a basis for each of these subspaces of the space P3 of cubic polynomials.
(a) The subspace of cubic polynomials p(x) such that p(7) = 0
(b) The subspace of polynomials p(x) such that p(7) = 0 and p(5) = 0
(c) The subspace of polynomials p(x) such that p(7) = 0, p(5) = 0, and p(3) = 0
(d) The space of polynomials p(x) such that p(7) = 0, p(5) = 0, p(3) = 0,
and p(1) = 0
1.25 We’ve seen that it is possible for a basis to remain a basis when it is reordered.
Must it remain a basis?
1.26 Can a basis contain a zero vector?
� 1.27 Let 〈 � β1, � β2, � β3〉 beabasisforavectorspace.
(a) Show that 〈c1� β1,c2� β2,c3� β3〉 is a basis when c1,c2,c3 �= 0. What happens
when at least one ci is 0?
(b) Prove that 〈�α1,�α2,�α3〉 is a basis where �αi = � β1 + � βi.
1.28 Give one more vector �v that will make each into a basis for the indicated
space.
� �
1
(a) 〈 ,�v〉 in R
1
2
� � � �
1 0
(b) 〈 1 , 1 ,�v〉 in R
0 0
3
(c) 〈x, 1+x 2 ,�v〉 in P2
� 1.29 Where 〈 � β1,..., � βn〉 is a basis, show that in this equation
c1 � β1 + ···+ ck � βk = ck+1 � βk+1 + ···+ cn � βn
each of the ci’s is zero. Generalize.
1.30 A basis contains some of the vectors from a vector space; can it contain them
all?
1.31 Theorem 1.12 shows that, with respect to a basis, every linear combination is
unique. If a subset is not a basis, can linear combinations be not unique? If so,
must they be?
� 1.32 A square matrix is symmetric if for all indices i and j, entryi, j equals entry
j, i.
(a) Find a basis for the vector space of symmetric 2×2 matrices.
(b) Find a basis for the space of symmetric 3×3 matrices.
(c) Find a basis for the space of symmetric n×n matrices.
� 1.33 We can show that every basis for R 3 contains the same number of vectors,
specifically, three of them.
(a) Show that no linearly independent subset of R 3 contains more than three
vectors.