1.4
Solutions
23
2. The matrix-vector product Ax product is not defined because the number of columns (1) in the 31
2
5
matrix 6 does not match the number of entries (2) in the vector .
1
1
5
6
5 12 15 3
2
3 = 2 4 3 3 = 8 + 9 = 1 , and
3
6
7

54
CHAPTER 1
Linear Equations in Linear Algebra
43. [M] Make v any one of the columns of A that is not in B and row reduce the augmented matrix [B v].
The calculations will show that the equation Bx = v is consistent, which means that v is a linear
combin

62
CHAPTER 1
1.9
Linear Equations in Linear Algebra
SOLUTIONS
Notes: This section is optional if you plan to treat linear transformations only lightly, but many instructors
will want to cover at least Theorem 10 and a few geometric examples. Exercises 15

1.6
1.6
Solutions
39
SOLUTIONS
1. Fill in the exchange table one column at a time. The entries in a column describe where a sector's output
goes. The decimal fractions in each column sum to 1.
Distribution of
Output From:
Goods
output
Services
input
.7
.

30
CHAPTER 1
Linear Equations in Linear Algebra
Note: Exercises 41 and 42 help to prepare for later work on the column space of a matrix. (See Section 2.9 or
4.6.) The Study Guide points out that these exercises depend on the following idea, not explicitl

8
CHAPTER 1
1.2
Linear Equations in Linear Algebra
SOLUTIONS
Notes: The key exercises are 120 and 2328. (Students should work at least four or five from Exercises
714, in preparation for Section 1.5.)
1. Reduced echelon form: a and b. Echelon form: d. Not

1.1
SOLUTIONS
Notes: The key exercises are 7 (or 11 or 12), 1922, and 25. For brevity, the symbols R1, R2, stand for
row 1 (or equation 1), row 2 (or equation 2), and so on. Additional notes are at the end of the section.
1.
x1 + 5 x2 = 7
2 x1 7 x2 = 5
1