Linear Algebra

130 Chapter 2. Vector Spaces
(a)
�
1
1
� � �
1 1
,
1 3
(b)
�
1
2
1
3
0
−3
� � �
1 1
4 , 0
−3 0
� 3.19 Find a basis for the row space of this matrix.
⎛
⎞
2 0 3 4
⎜0
⎝
3
1
1
1
0
−1⎟
2
⎠
1 0 −4 −1
� 3.20 Find � the rank�of each matrix. �
2 1 3
1 −1
�
2
(a) 1 −1 2 (b) 3 −3 6 (c)
1
�
0
0
0
3
�
0
−2 2 −4
(d) 0 0 0
0 0 0
� 3.21 Find a basis for the span of each set.
(a) { � 1 3 � , � −1 3 � , � 1 4 � , � 2 1 � }⊆M1×2
� � � � � �
1 3 1
(b) { 2 , 1 , −3 }⊆R
1 −1 −3
3
(c) {1+x, 1 − x 2 , 3+2x− x 2 }⊆P3
� � � � �
1 0 1 1 0 3 −1
(d) {
,
,
3 1 −1 2 1 4 −1
0
−1
�
−5
−9
�
1 3
�
2
5 1 1
6 4 3
}⊆M2×3
3.22 Which matrices have rank zero? Rank one?
� 3.23 Given a, b, c ∈ R, what choice of d will cause this matrix to have the rank of
one?
�
a
�
b
c d
3.24 Find the column rank of this matrix.
�
1 3 −1 5 0
�
4
2 0 1 0 4 1
3.25 Show that a linear system with at least one solution has at most one solution if
and only if the matrix of coefficients has rank equal to the number of its columns.
� 3.26 If a matrix is 5×9, which set must be dependent, its set of rows or its set of
columns?
3.27 Give an example to show that, despite that they have the same dimension,
the row space and column space of a matrix need not be equal. Are they ever
equal?
3.28 Show that the set {(1, −1, 2, −3), (1, 1, 2, 0), (3, −1, 6, −6)} does not have the
same span as {(1, 0, 1, 0), (0, 2, 0, 3)}. What, by the way, is the vector space?
� 3.29 Show that this set of column vectors
�� � �
d1 �� 3x +2y +4z = d1
d2 there are x, y, andzsuch that x − z = d2
d3
2x +2y +5z = d3
is a subspace of R 3 . Find a basis.