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a. Show that the line through vectors $\mathbf{p}$ and $\mathbf{q}$ in $\mathbb{R}^{n}$ may be written in the parametric form $\mathbf{x}=(1-t) \mathbf{p}+t \mathbf{q} .$ (Refer to the figure with Exercises 21 and 22 in Section $1.5 . )$b. The line segment from $\mathbf{p}$ to $\mathbf{q}$ is the set of points of the form $(1-t) \mathbf{p}+t \mathbf{q}$ for $0 \leq t \leq 1$ (as shown in the figure below). Show that a linear transformation $T$ maps this line segment onto a line segment or onto a single point.

A. $\boldsymbol{x}=(1-t) \boldsymbol{p}+t \boldsymbol{q}$B. See Solution

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 8

Introduction to Linear Transformations

Introduction to Matrices

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Okay, This question There are two parts A and B. So first we will solve a So the question is show that show that the line true vectors B and Q in are in. Maybe the return is maybe written is X equals two when my Nestea P plus take you Okay, so we know that it is only question is X equals toe p plus T Q minus B. And this can be written as p minus tp plus take you. And now we will take a peek. Woman then one dynasty be glass take you So this equation and the sufficient are same. So we have saw that. And now for be barred for be Bart. The ocean is the line segment The line segment segment from P to Q. Is the set off point off the form one minus D p plus? Did you four values off t will be theirs and it was 21 and great 20 questo zero and and we have to show that the linear transformation TV maps this line segment on the way line segment or onto a single point. Okay. For this we will take We will take X equals toe one minus D b plus Take you on t x Will be be when my next DP plus dick. You okay on? We can further sold at one minus. T dp plus de dick. You okay for the values off T this, Okay. And now we will further soul there. Okay, so this will be one way, and this will be the value off the X. Okay, so this has to answers. Okay. And the first one is first one is t X equals to one minus D d p plus t. Did you? So this is the proof that mhm the team maps this line segment on to a line segment. Okay. And if p equals took you here, then this will be DX equals toe one. My nasty d b plus de tp. Okay, then deep in the teepee, cancel on the final will be TV. And this is the This is map This maps on the single point only. Okay, so these two are the answers shall be over. Final answer. Thank you.

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