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 26E from Chapter 2.3 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 26E
Chapter:
Problem:
Explain why the columns of A2 span ℝn whenever the columns of an n × n matrix A are linearly independent.
Step-by-Step Solution
Given Information
We have to explain why the columns of $A^2$ span $R^n$ whenever the columns of A are linearly independent.
Step-1:
According to the invertible matrix theorem if A is an $n \times n$ matrix then matrix A is invertible if and only if columns of matrix A form a linearly independent set.
Step-2:
Since, the columns of A are linearly independent. matrix A is invertible. Hence square of the matrix is also invertible. For an invertible matrix, the columns of the matrix span $R^n$.
The columns of matrix $A^2$ span $R^n$
We have to explain why the columns of $A^2$ span $R^n$ whenever the columns of A are linearly independent.
Step-1:
According to the invertible matrix theorem if A is an $n \times n$ matrix then matrix A is invertible if and only if columns of matrix A form a linearly independent set.
Step-2:
Since, the columns of A are linearly independent. matrix A is invertible. Hence square of the matrix is also invertible. For an invertible matrix, the columns of the matrix span $R^n$.
The columns of matrix $A^2$ span $R^n$