Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A “by inspection.” Then find a basis of the image of A and a basis of the kernel of A.
16. $[\begin{matrix} 1 & - 2 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$

The redundant column vectors of matrix A $\vec{v_{2}} and \vec{v_{4}}$ .

The basis of the image of A = $\{[1 0 0], [0 1 0], [0 0 1]\}$

The basis of the kernel of A = $\{[2 1 0 0 0], [1 0 - 5 1 0]\}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Finding the redundant vectors

Let $\vec{v_{1}} = [1 0 0], \vec{v_{2}} = [- 2 0 0], \vec{v_{3}} = [0 1 0], \vec{v_{4}} = [- 1 5 0], \vec{v_{5}} = [0 0 1]$

Here we have $\vec{v_{2}} = 2 \vec{v_{1}} and \vec{v_{4}} = - \vec{v_{3}} + 5 \vec{v_{3}}$

$\Rightarrow \vec{v_{2}}$ are $\vec{v_{4}}$ redundant vectors and $\vec{v_{1}}, \vec{v_{3}} and \vec{v_{5}}$ are non-redundant vectors.

Step 2: Finding the basis of the image of A

The non-redundant column vectors of A form the basis of the image of A.

Since column vectors $\vec{v_{1}}, \vec{v_{3}} and \vec{v_{5}}$ non-redundant vectors of matrix A, thus the basis of the image A = $\{[1 0 0], [0 1 0], [0 0 1]\}$ i.e. .

Step 3: Finding the basis of the kernel of A

$\vec{v_{2}} and \vec{v_{4}}$ Since the vectors are redundant vectors such that

$\vec{v_{2}} = 2 \vec{v_{1}} and \vec{v_{4}} = - \vec{v_{1}} + 5 \vec{v_{3}} \Rightarrow 2 \vec{v_{1}} + \vec{v_{2}} + 0 and \vec{v_{1}} - 5 \vec{v_{3}} + \vec{v_{4}} = 0$

Thus the vectors in kernel of A is $\vec{w_{1}} = [2 1 0 0 0], \vec{w_{2}} = [1 0 - 5 1 0]$ .

Step 4: Final Answer

The redundant column vectors of matrix A are $\vec{v_{2}} and \vec{v_{4}}$ .

The basis of the image of A = $\{[1 0 0], [0 1 0], [0 0 1]\}$

The basis of the kernel of A = $\{[2 1 0 0 0], [1 0 - 5 1 0]\}$

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Linear Algebra With Applications

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Short Answer

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A “by inspection.” Then find a basis of the image of A and a basis of the kernel of A.
16. $[\begin{matrix} 1 & - 2 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$

Step by Step Solution

Step 1: Finding the redundant vectors

Step 2: Finding the basis of the image of A

Step 3: Finding the basis of the kernel of A

Step 4: Final Answer

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Linear Algebra With Applications

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Short Answer

Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A “by inspection.” Then find a basis of the image of A and a basis of the kernel of A.16.[1-20-100015000001]

Step by Step Solution

Step 1: Finding the redundant vectors

Step 2: Finding the basis of the image of A

Step 3: Finding the basis of the kernel of A

Step 4: Final Answer

Most popular questions for Math Textbooks

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Question: In Exercises 1 through 20, find the redundant column vectors of the given matrix A “by inspection.” Then find a basis of the image of A and a basis of the kernel of A.
16. $[\begin{matrix} 1 & - 2 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$