Q1.3 37E Find all vectors x鈫� such that ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q1.3 37E (page 36)

Find all vectors $\overset{鈫�}{x}$ such that $A \overset{鈫�}{x} = \overset{鈫�}{b}$ where $A = [\begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}] a n d \overset{鈫�}{b} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}]$ .

Short Answer

Expert verified

For $A = [\begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}] a n d \overset{鈫�}{b} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}]$ ,the vector, $\overset{鈫�}{x} = [\begin{matrix} 2 - 2 x_{2} \\ x_{2} \\ 1 \end{matrix}]$ , where $x_{2}$ is arbitrary.

Step by step solution

Consider the matrices.

If A is an $n 脳 m$ matrix with row vectors $\overset{鈫�}{蝇_{1}}, . . . \overset{鈫�}{蝇_{n}} a n d \overset{鈫�}{x}$ is a vector in role="math" localid="1664212679473" ${鈩�}^{m}$ then,

$A \overset{鈫�}{x} = [\begin{matrix} - {\overset{鈫�}{蝇}}_{1} - \\ 鈰� \\ - \overset{鈫�}{蝇_{n}} - \end{matrix}] \overset{鈫�}{x} [\begin{matrix} - {\overset{鈫�}{蝇}}_{1} . \overset{鈫�}{x} - \\ 鈰� \\ - \overset{鈫�}{蝇_{n}} . \overset{鈫�}{x} - \end{matrix}]$

The matrices are,

role="math" localid="1664212951470" $A = [\begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}] \overset{鈫�}{b} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}]$

The vector $\overset{鈫�}{x}$ is, role="math" localid="1664212989967" $\overset{鈫�}{x} = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}]$

Perform the multiplication operation.

Consider the matrices

$A \overset{鈫�}{x} = \overset{鈫�}{b} [\begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}] [x_{1} + 2 x_{2}] = 2 [x_{3}] = 1$

Therefore, the values are, $\overset{鈫�}{x} = [\begin{matrix} 2 - 2 x_{2} \\ x_{2} \\ 1 \end{matrix}]$ , here $x_{2}$ is arbitrary.

Hence, $\overset{鈫�}{x} = [\begin{matrix} 2 - 2 x_{2} \\ x_{2} \\ 1 \end{matrix}]$ where $x_{2}$ is arbitrary for $A = [\begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}] a n d \overset{鈫�}{b} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}]$ .

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91影视

Find all vectors $\overset{鈫�}{x}$ such that $A \overset{鈫�}{x} = \overset{鈫�}{b}$ where $A = [\begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}] a n d \overset{鈫�}{b} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}]$ .

Short Answer

Step by step solution

Consider the matrices.

Perform the multiplication operation.

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91影视

Find all vectors x鈫� such that Ax鈫�=b鈫� where A=[120001000]andb鈫�=[210].

Short Answer

Step by step solution

Consider the matrices.

Perform the multiplication operation.

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Find all vectors $\overset{鈫�}{x}$ such that $A \overset{鈫�}{x} = \overset{鈫�}{b}$ where $A = [\begin{matrix} 1 & 2 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}] a n d \overset{鈫�}{b} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}]$ .