17.) True or False? Write entire word TRUE or entire word FALSE in the space after each statement: Two points each: If you choose to not answer, you will receive ...

In Exercises 19 and $20,$ all vectors are in $\mathbb{R}^{n} .$ Mark each statement True or False. Justify each answer.
a. $\mathbf{u} \cdot \mathbf{v}-\mathbf{v} \cdot \mathbf{u}=0$
b. For any scalar $c,\|c \mathbf{v}\|=c\|\mathbf{v}\|$
c. If $\mathbf{x}$ is orthogonal to every vector in a subspace $W,$ then $\mathbf{x}$ $\quad$ is in $W^{\perp}$
d. If $\|\mathbf{u}\|^{2}+\|\mathbf{v}\|^{2}=\|\mathbf{u}+\mathbf{v}\|^{2}$ , then $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.
e. For an $m \times n$ matrix $A,$ vectors in the null space of $A$ are orthogonal to vectors in the row space of $A .$

In this question were given different statements on Told to determine if these statements are true or false. The first statement we have is V times vehicles too. Well, uh so is this statement Draw falls. This statement is true. Why is it true? Because we know that the length off a vector is equals to the square. Root off the entries in the victors. So we have V thank v. So if you square both sides, these statements is the same as the statesman. So it must be true. The next question we have or the next statements we have is for a scale Asi is supposed to see On what? Told to prove that this statement is true Falls Festival The statement is true. Why is it true? Because we know that Victor's have a distributive quality. So u times V is the same thing as V times You okay? Okay be you James V is equals to the also you times a week supposed to you into tv Whatever combination Off this off You beyonc mostly applied. We'll give you the same answer. So therefore these statements must be true. The next question is what told to prove that if you to be is equals to between U and V. Ah, Okay. Now on we know that for vectors to be auto go now they have to be perpendicular. So let's look at you on being perpendicular. This is you and this is the And this would be my nose v The distance from here to here from YouTube V has to be the Samos from you to minus B. So therefore, this statement is true. The next statement will give is in the square metrics if vectors in the face costume, I ought to go now. So basically, if vectors in Karume a, uh also go now two victors in No a in the square. My tricks This this man true. Ah, vectors in column A or to go Now Victor's in no way any square matrix festival we're gonna fest Define on a square matrix, for example, we have the square matrix 11 is there was zero So now we're gonna find no a which is equals to you square my tricks into x y So explosive why he goes to zero except the goes to minus Sorry. Why is it goes to minus eggs. So therefore X y it goes to if why is it goes to minus eggs? We can substitute minus eggs in place of why, Yeah, So this is gonna be, you know, is equal to one and minus one. So I'm gonna find the those products off A and more. It is one times one is one. There are times minus one is zero is close to one. So therefore it's not equals to zero. So therefore no a is no October now to any So therefore the statement is false Is not octagonal thio the last statement? Yeah, if vectors v to BP, I need subset of you. If X is auto, go now to V J for Jay Z goes to one then what told So? Therefore the the question says then X is the view Now what does this mean? This is cold on October now. Compliments. Thank you. So what is an octagonal compliment? The vector is also go now to every vector in a subspace w. Then the vector is said to be a not O'Connell compliments, which is the very definition. What told that X? What do the eggs is also go now to every vector in the subspace B J, which is Jay Z goes toe one. So therefore this statement is true. X is a octagonal.

Were given a statement and were asked to determine if the statement is true or false and to justify your answer. The statement is if a B and a C R parallel vectors, then the points A B and C are Colin ear. Well, suppose that a b this parallel to a C that these are vectors. Moreover, well, this implies that there is some scalar c such that vector A B is equal to some skater Multiple C times vector a c, of course, yeah, we know that vector a B is found by taking vector B and subtracting vector A just like vector A C is equal to vector C minus vector A and therefore we have that vector B minus factor A is equal to see times vector C minus vector A. In other words, we have that victor B minus factor a over vector C minus factor A is equal to a constant C. Of course. Vector B minus. Specter A over back there. See my inspector A magnitudes. Well, this implies that the slope of the line so called M is constant. The statement is true

Who asked if U. Is a unit factor. I usually know to help them with the hat. Um Then if that's the inter factor then our A. B. And C. The direction co signs of that Director. And that is true and why that is. Remember the direction co signs are the um Director um dotted with each of the directions divided by the magnitude of the vector provided by the magnitude of the direction vector. Or this is always one. And so in this case, um Because the this factor is a unit factor, this is also one. So what this says is that the co sign? The direction co sign with the X. Axis is you got it with X. You got it with my and that's just the component. Hey, so this is in fact this equals A. This equals B. In the secrecy. So if we do the same thing with the J vector, we get co signed a letter to the direction detection. Co signers you dotted with jay and that's B. And the direction. Third direction co sign is you gotta with K, and that just happens to be C. So if this is a unit vector, then its components on on the Cartesian basis, or the direction co signs with that basis.

So in again. Question. If we and be a non Jew directors, then the orthogonal production off Viane be its director. That is peddled. Toby reacted with this. A statement we have toe are we have to find the disconnect or not. So let's let's assume there is a vector the vector. And there is a rector be vector light. So if we find the production off director on be vector, then the production will be like this. Like, this is the vector, and this is be vectors off the production off we worked on. We will be, uh, in this direction, which is parallel to the direction off the reactor. Right. So the given courage statement ISC, uh, toe