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Explain why the columns of $A^{2}$ span $\mathbb{R}^{n}$ whenever the columns of $A$ are linearly independent.

Therefore, the columns of matrix $A^{2}$ span $\mathbb{R}^{n}$ .

Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

Campbell University

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

01:21

Suppose the columns of a m…

02:29

Suppose $A$ is an $m \time…

01:15

Let $\mathcal{B}=\left\{\m…

03:43

Explain why the columns of…

00:49

Suppose columns $1,3,5,$ a…

06:28

Let $A$ be an $m \times n$…

02:46

Suppose $A$ is a $3 \times…

10:34

Show that if the columns o…

If $L$ is $n \times n$ and…

01:18

Suppose $W$ is a subspace …

Okay, so in this problem, work, even new tricks, eh? Which column? Which columns? Already there. You know the independent. Oh. Which columns are you? Nearly independent. So what does this mean? Well, I over on our convertible Matrix Spear. Um, So what this means hey, is convertible. And there again by our incredible matrix fear. Um, there must exist. Matric. Uh, I'm back. And make sure you speak such that a times B his identity? No. Now we can see there a squared. So that is Hey, time's a So we consider it's cedar a times, eh? Times be times be And we consider this when we remember the matrix Multiplication is associate social social TV so we can calculate the time speedy inside first so that it's a times he times v will be I didn't hear So that's ight nd times be and I didn't he times a is just a can a time speeds identity. All right, so that means netting plies a squared that implies he squared ties b squared. He's I don't know. Furthermore, be square is in verse of a squared, so that means a square things is inevitable. And by our incredible matrix fear. Um, if we have a murder mystery stuff than a column of the squared must span our end. So a square conclusion is a square, you convertible and columns all those spin are in, so

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