Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

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.
20. $[\begin{matrix} 1 & 0 & 5 & 3 & - 3 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$

The redundant column vectors of matrix A are ${\overset{\leftrightarrow}{v}}_{2}, {\vec{v}}_{3} and {\vec{v}}_{4}$ .

The basis of the image of A = $\{[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 3 \\ 3 \\ 0 \\ 0 \end{matrix}]\}$ .

The basis of the kernel of A = role="math" localid="1659419993266" $\{[\begin{matrix} 0 \\ - 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 5 \\ 0 \\ - 1 \\ 0 \\ 0 \end{matrix}], \begin{matrix} [\begin{matrix} 4 \\ 0 \\ 0 \\ - 1 \\ \frac{1}{3} \end{matrix}] \end{matrix}\}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Finding the redundant vectors

Let ${\vec{v}}_{1} = [\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], {\vec{v}}_{2} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \end{matrix}], {\vec{v}}_{3} = [\begin{matrix} 5 \\ 0 \\ 0 \\ 0 \end{matrix}] {\vec{v}}_{4} = [\begin{matrix} 3 \\ 1 \\ 0 \\ 0 \end{matrix}] {\vec{v}}_{5} = [\begin{matrix} - 3 \\ 3 \\ 0 \\ 0 \end{matrix}]$

Here we have ${\vec{v}}_{2} = 0 ., {\vec{v}}_{1}, {\vec{v}}_{3} = 5 {\vec{v}}_{1} and {\vec{v}}_{4} = 4 {\vec{v}}_{1} + \frac{1}{3} {\vec{v}}_{5}$ .

$\Rightarrow {\vec{v}}_{2}, {\vec{v}}_{3} and {\vec{v}}_{4}$ are redundant vectors and ${\vec{v}}_{1} and {\vec{v}}_{5}$ 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 ${\vec{v}}_{1} and {\vec{v}}_{5}$ are non-redundant vectors of matrix A, thus the basis of the image A = $\{{\vec{v}}_{1}, {\vec{v}}_{5}\} i . e \{[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 3 \\ 3 \\ 0 \\ 0 \end{matrix}]\}$ .

Step 3: Finding the basis of the kernel of A

Since the vectors ${\vec{v}}_{2}, {\vec{v}}_{3} and {\vec{v}}_{4}$ are redundant vectors such that

${\vec{v}}_{2} = 0 . {\vec{v}}_{1}, {\vec{v}}_{3} = 5 {\vec{v}}_{1} a n d {\vec{v}}_{4} = 4 {\vec{v}}_{1} + \frac{1}{3} {\vec{v}}_{5} \Rightarrow 0 . {\vec{v}}_{1} - {\vec{v}}_{2} = 0, 5 {\vec{v}}_{1} - {\vec{v}}_{3} = 0 a n d 4 {\vec{v}}_{1} - {\vec{v}}_{4} + \frac{1}{3} {\vec{V}}_{5} = 0$

Thus the vectors in the kernel of A are ${\vec{w}}_{1} = [\begin{matrix} 0 \\ - 1 \\ 0 \\ 0 \\ 0 \end{matrix}], {\vec{w}}_{2} = [\begin{matrix} 5 \\ 0 \\ - 1 \\ 0 \\ 0 \end{matrix}], {\vec{w}}_{3} = \begin{matrix} [\begin{matrix} 4 \\ 0 \\ 0 \\ - 1 \\ \frac{1}{3} \end{matrix}] \end{matrix}$ .

Step 4: Final Answer

The redundant column vectors of matrix A are ${\vec{v}}_{2}, {\vec{v}}_{3} and {\vec{v}}_{4}$ .

The basis of the image of A = $A = \{[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} - 3 \\ 3 \\ 0 \\ 0 \end{matrix}]\}$ .

The basis of the kernel of A = $\{[\begin{matrix} 0 \\ - 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 5 \\ 0 \\ - 1 \\ 0 \\ 0 \end{matrix}], \begin{matrix} [\begin{matrix} 4 \\ 0 \\ 0 \\ - 1 \\ \frac{1}{3} \end{matrix}] \end{matrix}\}$ .

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Linear Algebra With Applications

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Short Answer

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.
20. $[\begin{matrix} 1 & 0 & 5 & 3 & - 3 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$

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

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.20. [1053-3000130000000000]

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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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.
20. $[\begin{matrix} 1 & 0 & 5 & 3 & - 3 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$