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Let $T : \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ be a linear transformation, and let $\mathbf{p}$ be a vector and $S$ a set in $\mathbb{R}^{m} .$ Show that the image of $\mathbf{p}+S$ under $T$ is the translated set $T(\mathbf{p})+T(S)$ in $\mathbb{R}^{n} .$

$T(\mathbf{p})+T(S)$

Algebra

Chapter 3

Determinants

Section 3

Cramer’s Rule, Volume, and Linear Transformations

Introduction to Matrices

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Lectures

01:32

In mathematics, the absolu…

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Let $f : \mathbb{R}^{n} \r…

04:12

Let $S$ be an affine subse…

02:40

Suppose vectors $\mathbf{v…

02:02

Let $T_{1}: M_{n}(\mathbb{…

06:35

Let $V$ and $W$ be vector …

06:54

Let $T$ be a linear transf…

03:38

Let $T_{1}: V \rightarrow …

03:39

Let $T$ be a one-to-one li…

04:32

Let $V$ be a real inner pr…

03:34

in this video, we're gonna be solving problems over 26 of section 2.3, which is based on linear independence. Ah, transformation or not, linear independence. I'm sorry. It's based on determinants and Cramer's rule and, uh, linear transformations and volumes. So here we're given that Tia's transformation, um from our himto are in and we know that p is a vector and s is a set in r r m s as is a set in, um our, um So let we can let this is a proof so we can let the vector v be any vector any vector from the set s which is on our M. And by definition, we know that people s is a set of all vectors because s a set of all vectors, um, better of the form p plus V that we did find your love. Uh and we know that V isn't s so applying t to ah, typical vector and people cess. We have, um t and we apply people. Savi innit would give us tp plus t v. And this is this vector is in the said denoted by t plus s her teepee plus t s because we know that he is then s This proves that t maps the set people says onto the set T p plus t s. So this proves that, um, t as t maps people us us onto t p plus t s was what the initial problems asking which stated that we need to show that the image of people says under T a translated set t put people's t s in rn. So there's just a simple proof so shit.

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