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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q15E (page 35)

Compute the products Ax in Exercises 13 through 15 using
paper and pencil. In each case, compute the product
two ways: in terms of the columns of A and in terms of the rows of A.
15.
$[1 2 3 4] [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]$

Short Answer

Expert verified

The dot product $A x^{鈫�}$ of $[1 2 3 4] [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]$ is [70].

Step by step solution

01

Step 1:

Find $A x .$ in terms of rows of A.

$[1 2 3 4] [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]$

$鈬� [(1) (5) + (2) (6) + (3) (7) + (4) (8)] 鈬� [5 + 12 + 21 + 32] 鈬� [70]$

02

Step 2:

Find $A x .$ in terms of columns of A.

$[1 2 3 4] [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]$

$鈬� 5 [1] + 6 [2] + 7 [3] + 8 [4] 鈬� [5 + 12 + 21 + 32] [70]$

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Most popular questions from this chapter

Solve the linear system of equations. You may use technology.

$| \begin{matrix} 3 x_1 + 6 x_2 + 9 x_3 + 5 x_4 + 25 x_5 & = 53 \\ 7 x_1 + 14 x_2 + 21 x_3 + 9 x_4 + 53 x_5 & = 105 \\ - 4 x_1 - 8 x_2 - 12 x_3 + 5 x_4 - 10 x_5 & = 11 \end{matrix} |$

10: In Exercises 1 through 12, find all solutions of the equations

with paper and pencil using Gauss鈥揓ordan elimination.

Show all your work.

$| \begin{matrix} 4 x_{1} + 3 x_{2} - 2 x_{3} - x_{4} = 4 \\ 5 x_{1} + 4 x_{2} - 3 x_{3} - x_{4} = 4 \\ - 2 x_{1} - 2 x_{2} - x_{3} - 2 x_{4} = - 3 \\ 11 x_{1} + 6 x_{2} + 4 x_{3} - x_{4} = 11 \end{matrix} |$

Question: Give a geometrical description of the set of all vectors $\overset{鈬赌}{u} i n R^{2}$ such that $\overset{鈬赌}{u} . \overset{鈬赌}{v} = \overset{鈬赌}{u} . \overset{鈬赌}{w}$ .

Question:A linear system of the form $A \overset{鈫�}{x} = 0$ is called homogeneous. Justify the following facts:

a.All homogeneous systems are consistent.

b.A homogeneous system with fewer equations than unknowns has infinitely many solutions.

c.If $\overset{鈫�}{x_{1}} and \overset{鈫�}{x_{2}}$ are solutions of the homogeneous system $A \overset{鈫�}{x} = 0$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{2}}$ is a solution as well.

d.If $\overset{鈫�}{x}$ is a solution of the homogeneous system $A \overset{鈫�}{x} = 0$ andkis an arbitrary constant, then $k \overset{鈫�}{x}$ is a solution as well.

For which values of a, b, c, d, and eis the following matrix in reduced row-echelon form?

$A = [\begin{matrix} 0 & a & 2 & 1 & b \\ 0 & 0 & 0 & c & d \\ 0 & 0 & e & 0 & 0 \end{matrix}]$

See all solutions

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