Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Question: Consider linearly independent vectors $\vec{v_{1}}, \vec{v_{2}}, . . . ., \vec{v_{m}}$ in $ℝ^{n}$ and let A be an invertible $m \times m$ matrix. Are the columns of the following matrix linearly independent?
$[\begin{matrix} \frac{I}{V_{1}} & \frac{I}{V_{2}} . . . & \frac{I}{V_{m}} \\ I & I & I \end{matrix}] A$

Yes, the columns of the matrix linearly independent.

$[\begin{matrix} \frac{I}{V_{1}} & \frac{I}{V_{2}} . . . & \frac{I}{V_{m}} \\ I & I & I \end{matrix}] A$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of Linearly independent

Linearly independent is defined as the property of a set having no linear combination of its elements equal to zero when the coefficients are taken from a given set unless the coefficient of each element is zero.

Step 2: Prove the vectors are linearly independent

$\vec{v_{1}}, \vec{v_{2}}, . . . ., \vec{v_{m}}$ are linearly independent vectors.

We are given a $m \times m$ matrix A that is invertible.

Let’s say that

$B = [\begin{matrix} \frac{I}{V_{1}} & \frac{I}{V_{2}} . . . & \frac{I}{V_{m}} \\ I & I & I \end{matrix}]$

Since they are independent we know that ker B = 0

Therefore kerBA = 0 so columns of AB are linearly independent.

Step 3: The final answer

Yes, the columns of the matrix are linearly independent.

$[\begin{matrix} \frac{I}{V_{1}} & \frac{I}{V_{2}} . . . & \frac{I}{V_{m}} \\ I & I & I \end{matrix}] A$

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Linear Algebra With Applications

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Step by Step Solution

Step 1: Definition of Linearly independent

Step 2: Prove the vectors are linearly independent

Step 3: The final answer

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Linear Algebra With Applications

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Short Answer

Question: Consider linearly independent vectors v1→,v2→,....,vm→ in ℝn and let A be an invertible m×mmatrix. Are the columns of the following matrix linearly independent?[IV1IV2...IVmIII]A

Step by Step Solution

Step 1: Definition of Linearly independent

Step 2: Prove the vectors are linearly independent

Step 3: The final answer

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