Q44E Question: Consider linearly inde... [FREE SOLUTION]

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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q44E (page 132)

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

Short Answer

Expert verified

Yes, the columns of the matrix linearly independent.

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

Step by step solution

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.

Prove the vectors are linearly independent

$\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, . . . ., \overset{鈫�}{v_{m}}$ are linearly independent vectors.

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

Let鈥檚 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.

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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Short Answer

Step by step solution

Definition of Linearly independent

Prove the vectors are linearly independent

The final answer

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91影视

Question: Consider linearly independent vectors v1鈫�,v2鈫�,....,vm鈫�in 鈩�nand let A be an invertible m脳mmatrix. Are the columns of the following matrix linearly independent?[IV1IV2...IVmIII]A

Short Answer

Step by step solution

Definition of Linearly independent

Prove the vectors are linearly independent

The final answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Geometry

Calculus

Decision Maths

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.