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Let $T : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be an invertible linear transformation. Explain why $T$ is both one-to-one and onto $\mathbb{R}^{n} .$ Use equations (1) and $(2) .$ Then give a second explanation using one or more theorems.

one to oneontoonto

Algebra

Chapter 2

Matrix Algebra

Section 3

Characterizations of Invertible Matrices

Introduction to Matrices

Oregon State University

McMaster University

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

05:13

Suppose $T$ and $S$ satisf…

02:04

Suppose $A$ is $n \times n…

02:13

An invertible linear trans…

01:04

04:28

Let $T : \mathbb{R}^{n} \r…

02:02

Let $T_{1}: M_{n}(\mathbb{…

02:27

Let $S : \mathbb{R}^{p} \r…

03:50

Suppose $A, B,$ and $X$ ar…

03:35

If $T: V \rightarrow W$ is…

05:20

In Exercises 33 and $34, T…

in problem 35 If we have tea and in veritable linear transformation, it's an in vertebral. Then you're transformation. We want to explain why he is both Oneto one on gone to our end using equations one and two. Because he is a linear, veritable linear transformation, it means it's a transformation. For example, this system X equals six we want when we want to blow it by t we're t is the inverse off T is the inverse off the which means we have the equation. One is given here and the second equation is 80 x equals x no for the first part of the problem for one toe one proof we want to prove that the X equals zero has the only reveal solution. We can assume X as any victor Such is that Z for example? Then we have TV equals zero demand by by V equals zero and boy applying the embers off t. We must apply both sides by a from left. Then we have a T V equals a multiplied boy zero that she gives zero. We can see that we have a TV, the left hand side and the left hand side is very similar to the equation, too. Where e t multiplied by a victor gives the same victor. Then we have the left hand side equals TV from Equation two and from the right hand side, it equals zero. This means this equation has only the trivial solution. We have proven that T V equals zero because we will be only equals zero. We will not be in a victor. V will be all those zero, which means the X has only trivial solution. And this proves in the first part of the problem for the second birth on to are in we want to prove that t multiplied by X equals B as at least one nontrivial solution. As at least one solution, you can prove this the same way as 1 to 1. Bye, I think I mean, you have t want to blow it by any victor. For example, W example w equals B. We want to prove that be has a solution. Is is not always be zero solution like this. We use equation too and multiply both sides from left by the inverse of T, which is a then we have 80 w equals a B and from equation, too. We have 80 multiply by Any Victor's gives the victor, then w equals the metrics. A multiply it by affected b will be is the non zero victor, which makes the right hand side, is down zero and has a solution. Which means that this equation gives a solution X equals w. And this proves the second board of the problem. And we have from fear, um, it we can prove that if mhm is in vertebral metrics, then it's a transformation. For example, T is 1 to 1 1 to 1 and it maps are in home to our in from the in vertebral metrics theory, and this is the final answer of all broken.

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In mathematics, the absolute value or modulus |x| of a real number x is its …

Suppose $T$ and $S$ satisfy the invertibility equations $(1)$ and (2), where…

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