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In Exercises 21 and $22,$ mark each statement True or False. Justify each answer.a. A linear transformation from $\mathbb{R}$ to $\mathbb{R}^{n}$ is called a linear functional.b. If $f$ is a linear functional defined on $\mathbb{R}^{n},$ then there exists a real number $k$ such that $f(\mathbf{x})=k \mathbf{x}$ for all $\mathbf{x}$ in $\mathbb{R}^{n}$ .c. If a hyperplane strictly separates sets $A$ and $B,$ then $A \cap B=\varnothing$ .d. If $A$ and $B$ are closed convex sets and $A \cap B=\varnothing$ , then there exists a hyperplane that strictly separates $A$ and $B$ .

a) $\mathrm{F}$b) $\mathrm{F}$c) $\mathrm{T}$d) $\mathrm{F}$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 4

Hyperplanes

Vectors

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okay, In this question, we will answer with a Siri's off truth and false is so a hey, if Israel number and F is a non zero linear functional than FT is a hard time in our So this diamond is true If you go to the textbook on page 4644464 The statement after the question three say this. Okay, given any vector in on any rule number, date, set, end, times X and secretive Gate is a high plane. This is false. Um, the victor in must be nausea. So Paige, six for six number Our Question three s a c if a and B a non empty. I'm not empty district sets such that a is compact and being close. Then there exists a high plane that's strictly separates A and B. This is false. The sets must also be convex. Well, they must be convex. So the scene Why, that's just crap. Very simple example. So well, over this set as a so this is both close and compact. Then we can create a set B, which is just a line. So this line B, this is a B is close So now we have a set which is strictly coast and compact. But there is no line. There's a hyper thing which week, which, with we can separate these two, there's no way do you if there exists 100 times such that taste of usted please separate two sets A and B in convicts of eight, capped by convicts of these non empty. This is false because some other hard thing, um, on the I could play cancer right then can't sit right them.

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