Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find a $n \times n$ matrix A such that $A \vec{x} = 3 \vec{x}$ for all $\vec{x}$ in $R^{n}$ .

Matrix A will be $3 I$ , where $I$ identity matrix of order is $n \times n$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step by step Explanation: Step1: System of equation

We have given that $A \vec{x} = 3 \vec{x}$ for $\vec{x}$ all in $R^{n}$ .

Which implies that $\vec{x}$ is $n \times 1$ matrix vector.

Step2: Linear Transformation

A transformation from $R^{m} \to R^{n}$ is said to be linear if the following condition holds:

Identity of $R^{m}$ should be mapped to identity of $R^{n}$ .
$T (a + b) = T (a) + T (b)$ For all $a, b \in R^{m}$ .
$T (c a) = c T (a)$ Where c is any scalar and $a \in R^{m}$ .

Step3: Simplification

We have $A \vec{x} = 3 \vec{x}$ . We can write it as

$A \vec{x} - 3 \vec{x} = 0 (A - 3) \vec{x} = 0$

Here, we have taken the vector x is non-zero.

This implies that

$\begin{array}{rcl} A - 3 & = & 0 \\ A & = & 3 I \end{array}$

Where $I$ is identity matrix of order $n \times n$ .

Thus, matrix A will be $3 I$ , where identity matrix of order is $n \times n$ .

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Linear Algebra With Applications

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Short Answer

Find a $n \times n$ matrix A such that $A \vec{x} = 3 \vec{x}$ for all $\vec{x}$ in $R^{n}$ .

Step by Step Solution

Step by step Explanation: Step1: System of equation

Step2: Linear Transformation

Step3: Simplification

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Linear Algebra With Applications

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Find a n×n matrix A such that Ax→=3x→ for all x→ in Rn.

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Find a $n \times n$ matrix A such that $A \vec{x} = 3 \vec{x}$ for all $\vec{x}$ in $R^{n}$ .