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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: Q3.4-18E (page 1)

In Exercises1 through 18, determine whether the vector $\overset{鈫�}{x}$ is in the span V of the vectors $\overset{鈬赌}{v_{1}},,,, \overset{鈬赌}{v_{m}}$ (proceed 鈥渂y inspection鈥� if possible, and use the reduced row-echelon form if necessary). If $\overset{鈫�}{x}$ is in V, find the coordinates of $\overset{鈫�}{x}$ with respect to the basis role="math" localid="1664193712120" $I = \overset{鈬赌}{v_{1}},,,, \overset{鈬赌}{v_{m}}$ of V, and write the coordinate vector $[\overset{鈬赌}{x}] I$ .
role="math" localid="1664193998673" $\overset{鈬赌}{x} = [\begin{matrix} 5 \\ 4 \\ 3 \\ 2 \end{matrix}]; \overset{鈬赌}{v_{1}} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 1 \end{matrix}]; \overset{鈬赌}{v_{2}} = [\begin{matrix} 0 \\ 1 \\ 1 \\ 0 \end{matrix}]; \overset{鈬赌}{v_{3}} = [\begin{matrix} 0 \\ - 1 \\ 0 \\ 1 \end{matrix}]$

Short Answer

Expert verified

As the coordinates of the vector are not available, so, $\overset{鈬赌}{x} I$ isn鈥檛 in V.

Step by step solution

01

Consider the vectors.

The vectors are,

$\overset{鈬赌}{x} = [\begin{matrix} 5 \\ 4 \\ 3 \\ 2 \end{matrix}]; \overset{鈬赌}{v_{1}} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 1 \end{matrix}]; \overset{鈬赌}{v_{2}} = [\begin{matrix} 0 \\ 1 \\ 1 \\ 0 \end{matrix}]; \overset{鈬赌}{v_{3}} = [\begin{matrix} 0 \\ - 1 \\ 0 \\ 1 \end{matrix}]$

02

Check for the coordinates of the vector.

The formula is, $\overset{鈬赌}{x} = c_{1} \overset{鈬赌}{v_{1}} + c_{2} \overset{鈬赌}{v_{2}}$ .

Consider the matrix form and perform rrref,

$r r e f [\begin{matrix} 1 & 0 & 0 & 5 \\ 1 & 1 & - 1 & 4 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 2 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]$

03

Final answer.

As the coordinates of the vector are not available, so, $\overset{鈬赌}{x} I$ isn鈥檛 in V .

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Most popular questions from this chapter

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.

$| \begin{matrix} x_{4} + 2 x_{5} - x_{6} = 2 \\ x_{1} + 2 x_{2} + x_{5} - x_{6} = 0 \\ x_{1} + 2 x_{2} + 2 x_{3 -} x_{5} - x_{6} = 2 \end{matrix} |$

The reduced row-echelon forms of the augmented matrices of three systems are given here. How many solutions does each system have?

$a . [\begin{matrix} 1 & 0 & 2 & | \\ 0 & 1 & 3 & | \\ 0 & 0 & 0 & | \end{matrix} \begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] b . [\begin{matrix} 1 & 0 & | & 5 \\ 0 & 1 & | & 6 \end{matrix}] c . [\begin{matrix} 0 & 1 & 0 & | & 2 \\ 0 & 0 & 1 & | & 3 \end{matrix}]$

a. Find $(\begin{matrix} 2 & 3 \\ 4 & 5 \\ 6 & 7 \end{matrix}) + (\begin{matrix} 7 & 5 \\ 3 & 1 \\ 0 & - 1 \end{matrix})$

b. Find $9 (\begin{matrix} 1 & - 1 & 2 \\ 3 & 4 & 5 \end{matrix})$

Show that any positive definite matrix A can be written as, where B is a positive definite matrix.

Compute the products Axin 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.

14.

$[\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{matrix}] [\begin{matrix} - 1 \\ 2 \\ 1 \end{matrix}]$

See all solutions

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