Linear Algebra

Section IV. Matrix Operations 219
We have now seen how the representation of the composition of two linear
maps is derived from the representations of the two maps. We have called
the combination the product of the two matrices. This operation is extremely
important. Before we go on to study how to represent the inverse of a linear
map, we will explore it some more in the next subsection.
Exercises
� 2.14 Compute, or state ‘not defined’.
� �� � �
3 1 0 5
1 1
(a)
(b)
−4 2 0 0.5
4 0
�
−1
3
� � �
2 −7
(c)
7 4
2
3
3
�
−1 −1
1 1
1 1
� �
1 0 5 �
5
−1 1 1 (d)
3
3 8 4
��
2 −1
1 3
�
2
−5
� 2.15 Where
A =
� �
1 −1
2 0
B =
� �
5 2
4 4
C =
�
−2
�
3
−4 1
compute or state ‘not defined’.
(a) AB (b) (AB)C (c) BC (d) A(BC)
2.16 Which products are defined?
(a) 3×2 times2×3 (b) 2×3 times 3 ×2 (c) 2×2 times3×3
(d) 3×3 times 2×2
� 2.17 Give the size of the product or state ‘not defined’.
(a) a2×3 matrix times a 3×1 matrix
(b) a1×12 matrix times a 12×1 matrix
(c) a2×3 matrix times a 2×1 matrix
(d) a2×2 matrix times a 2×2 matrix
� 2.18 Find the system of equations resulting from starting with
h1,1x1 + h1,2x2 + h1,3x3 = d1
h2,1x1 + h2,2x2 + h2,3x3 = d2
and making this change of variable (i.e., substitution).
x1 = g1,1y1 + g1,2y2
x2 = g2,1y1 + g2,2y2
x3 = g3,1y1 + g3,2y2
2.19 As Definition 2.3 points out, the matrix product operation generalizes the dot
product. Is the dot product of a 1×n row vector and a n×1 column vector the
same as their matrix-multiplicative product?
� 2.20 Represent the derivative map on Pn with respect to B,B where B is the
natural basis 〈1,x,... ,x n 〉. Show that the product of this matrix with itself is
defined; what the map does it represent?
2.21 Show that composition of linear transformations on R 1 is commutative. Is
this true for any one-dimensional space?
2.22 Why is matrix multiplication not defined as entry-wise multiplication? That
would be easier, and commutative too.
� 2.23 (a) Prove that H p H q = H p+q and (H p ) q = H pq for positive integers p, q.
(b) Prove that (rH) p = r p · H p for any positive integer p and scalar r ∈ R.