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Linear Algebra With Applications
Found in: Page 132
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

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


Yes, the columns of the matrix linearly independent.


See the step by step solution

Step by Step Solution

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

v1,v2,....,vm are linearly independent vectors.

We are given a m×m matrix A that is invertible.

Let’s say that


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.


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