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Let $T$ be a linear transformation that maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{n} .$ Show that $T^{-1}$ exists and maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{n} .$ Is $T^{-1}$ also one-to-one?

As the matrix $A^{-1}$ is the standard matrix for $T^{-1},$ the linear transformation $T^{-1}$ is one-to-oneand maps $\mathbb{R}^{n}$ onto $\mathbb{R}^{n}$ .

Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

Missouri State University

Campbell University

Oregon State University

Harvey Mudd College

Lectures

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Okay, so in this problem, we're given a linger transformation. Ron are in to our end. Okay, Now, the first question we need to answer is that whether whether verse exists, so you answer that question. The first note is that the minute information and B he noted as eight times X. So, um, by our definition, oh, onto so for any Why in rn there's always exists an axe from our end. Such that tee off acts was why so t up acts? It's just a few times acts just to hear, because here extra, it's the East available. I just inundated the available on the inside, but he actually is just a the mapping of acts under the transformation of tea. So it's just a T ax will be, Yes, because why so equivalent? Believe that means that the young to off this inner transformation implies the consistency off this in your system. So that means the system is consistent, sister. Okay, so by our bye convertible function convertible matrix theory, I I m t a is in vertical, so let's denote be to be the uber self hate. So we can find we can find a, um Lena Transformation. Asked which is staying first up t to be a inverse times axe and we substitute a inverse by. That will be be time, sex. So this is our, uh this is our universe off team. So ask. It's the inverse off t. So t is convertible, so that means t to universe exists. Okay, next question me to answer is whether the map in is on two. Um, to answer this question, we have to We still need to use a You see a bird convertible. Mac matrix. Incredible. Mission State here. Um, so we first consider the inverse up T, which is the axe. Now we follow the definition of round two. We just used a definition of onto to check whether this this new in your transformation will satisfy the definition of onto it does. Then we can say that is that is in your Central Asia, and he's on two. So take any why from our end. So what does there exist? Who, uh, that still exist and acts that makes as self eggs. He was white, your ex? And why are all victors Okay? So since the, um, the map enough x under the leaner transformation ass will be just We'll just be the p X. So this is saying as he acts, it was Why? So that is equivalent to say, um whether this linear system is consistent, whether the he's consistent. Well, the answer is yes, because we know he's in veritable. And since bees in veritable, we can apply over, he's a bit of a matrix here. Um, so that means that tells us B X equals why is consistent. So bye, I am t he actually was. Why is consistent? So that means there exists such an axe to make is to make a to make ass off acts. Because why? So that implies. Asked is too. So we're done. All right. Last question we need to answer is that whether the Lena transformation off in burst at his chambers, which is asked in this case, is 1 to 1. Well, this can't be. This can't be done. Just directly follow the convertible confirmations theorem because we know if B is in veritable and bye convertible matrix here on, we can at ease hards f just Do you know this part of my part? Off fingers? Incredible matrix the room. Um, we know that X two. Yeah, eggs is 1 to 1. And this is exactly our mapping ass are in to our end. So that means our as that means the transformation as he's 1 to 1. So we're done.

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