Linear Algebra, 2020a

242 Chapter Three. Maps Between Spaces
̌ 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 times 2×3 (b) 2×3 times 3×2 (c) 2×2 times 3×3
(d) 3×3 times 2×2
̌ 2.17 Give the size of the product or state “not defined”.
(a) a 2×3 matrix times a 3×1 matrix
(b) a 1×12 matrix times a 12×1 matrix
(c) a 2×3 matrix times a 2×1 matrix
(d) a 2×2 matrix times a 2×2 matrix
̌ 2.18 Find the system of equations resulting from starting with
h 1,1 x 1 + h 1,2 x 2 + h 1,3 x 3 = d 1
h 2,1 x 1 + h 2,2 x 2 + h 2,3 x 3 = d 2
and making this change of variable (i.e., substitution).
x 1 = g 1,1 y 1 + g 1,2 y 2
x 2 = g 2,1 y 1 + g 2,2 y 2
x 3 = g 3,1 y 1 + g 3,2 y 2
̌ 2.19 Consider the two linear functions h: R 3 → P 2 and g: P 2 → M 2×2 given as here.
⎛ ⎞
a
( )
⎝b⎠ p p− 2q
↦→ (a + b)x 2 +(2a + 2b)x + c px 2 + qx + r ↦→
q 0
c
Use these bases for the spaces.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 0
B = 〈 ⎝1⎠ , ⎝1⎠ , ⎝0⎠〉 C = 〈1 + x, 1 − x, x 2 〉
1 1 1
( ) ( ) ( ) ( )
1 0 0 2 0 0 0 0
D = 〈 , , , 〉
0 0 0 0 3 0 0 4
(a) Give the formula for the composition map g ◦ h: R 3 → M 2×2 derived directly
from the above definition.
(b) Represent h and g with respect to the appropriate bases.
(c) Represent the map g ◦ h computed in the first part with respect to the
appropriate bases.
(d) Check that the product of the two matrices from the second part is the matrix
from the third part.
2.20 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?
̌ 2.21 Represent the derivative map on P n 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 map does it represent?