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Show that if $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a basis for $\mathbb{R}^{3},$ then aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is the plane through $\mathbf{v}_{1}, \mathbf{v}_{2},$ and $\mathbf{v}_{3} .$

Consider that the set $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a basis for $\mathbb{R}^{3}$ .To prove that aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is the plane in $\mathbb{R}^{3}$ through $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3},$ use the following resultthe set aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a plane in $\mathbb{R}^{3[i]}$ and only if $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is linearly independentSince, the set $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is a basis for $\mathbb{R}^{3}$ , i. I' is linearly independent.So, from the above result aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$ is the plane in $\mathbb{R}^{3}$ through $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$ .

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 1

Affine Combinations

Vectors

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now for this question. Suppose V one v. TSU victory is a busies for our three now from problem for exercise That scene we of that's w equals a spon Vito Miners V want v three miners. V one is a blaine in all three. No, we know that this sets w plus v Juan is a blend of you translated because you're a DMV you want It's translated for view on this plane. This plane that is W plus V one would be Parlow will be parallel, so w and at the same time it will contain the one. Now, by definition, we know that ve to manners V one is in w right because the bees goes to spawn of days. So we are vey to minors. V one is in w This would imply that's V two minors view on, which is in W plus v Juan, which is the cost of itsu would be in w plus viewer because this is in W. Then this is V Juan. So this would be this is in w the want similarly so let's go about two dots. So this implies that we can just see Okay, I have it. Yes, so that implies that v two easy w plus view and that's what I wanted. I lights. So we have that. Similarly, similarly by definition, victory Miners v. Juan is in w. So this implies that Victory Miners v. Juan, plus V Juan isn't w and plus view on which is actually 1/4 just victory because V one were constant. This one this is in w plus v want. So this implies that's w close v one. No, Molly, it's contents. Everyone we've established that sits contains the one we've showed its components V two. On that we just showed that it also contains victory. Now this implies that's by the definition off w blows v Juan on dhe. If you also look at the serum one in your textbook, this would imply dots. I find off the Juan V to victory is the plane w blows V Juan dots. Contains v want VD

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