Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 22E from Chapter 2.1 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 22E
Chapter:
Problem:
Show that if the columns of B are linearly dependent, then so are the columns of AB.
Step-by-Step Solution
Given Information
We are given that A and B are two matrices such that columns of B are linearly dependent. We have to prove that columns of AB are also linearly dependent.
Step 1:
If the columns of matrix B are linearly dependent, then the system of equations given by BX=0 should have a solution.\[BX = 0\]Multiply both sides with matrix A. \[\begin{array}{l}A\left( {BX} \right) = A \times 0\\\left( {AB} \right)X = 0\end{array}\]Thus columns of AB must also be linearly dependent.
We are given that A and B are two matrices such that columns of B are linearly dependent. We have to prove that columns of AB are also linearly dependent.
Step 1:
If the columns of matrix B are linearly dependent, then the system of equations given by BX=0 should have a solution.\[BX = 0\]Multiply both sides with matrix A. \[\begin{array}{l}A\left( {BX} \right) = A \times 0\\\left( {AB} \right)X = 0\end{array}\]Thus columns of AB must also be linearly dependent.