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Let $\mathbf{v} \in \mathbb{R}^{n}$ and let $k \in \mathbb{R} .$ Prove that $S=\left\{\mathbf{x} \in \mathbb{R}^{n} : \mathbf{x} \cdot \mathbf{v}=k\right\}$ is an affine subset of $\mathbb{R}^{n}$ .

Therefore, is a subspace of the vector space.If satisfies, then is a translate of the subspace, so the set $S$ is a flat in.So, the set is an affine subset of.

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 1

Affine Combinations

Vectors

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okay for this question we have Let's V b, r N and Kay in our we want to show that one toe shoe. That's s which is, of course, tow X In our end. Given that ex dot v equals two K, these are fine subsets. Both are in. So do this by definition on page four nights warn we have to sure you have to show that one minus tse times p lost. See, Time's Cube is in s full p com acute in s. So this is in dilemma rights. It's much more elaborate for peek A McCue in S on DDE. Seeing is any real number. So now let's become a Q B s then, because they're in s on by definition of s, that implies that p dot v equals two K and similarly q dot v equals two K. Now, let's see be any real number, be any real number. Then we have that want minus c more supply by P. Lost See most like you dot v. We need to show that this is the coast. Okay, this is the way we can show that this thing is an s. So the show these we need to show that dese. Is it war too? Two. No. One minus C Times People lost cq dots V will be equal to one minus C p dots V plus C Q Dots Ving Now beat us. V is K So I have one minus tse multiply by cabe los que times That visa's okay, That's plus C k. When I saw this, this is called you K minus C k lost CK on. This is a course to K. So this implies that because we have Kay, this implies that s is are fine subsets off our end.

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