in-exercises-16-and-17-mark-each-statement-true-or-false-justify-each-answer-a-a-cube-in-mathbbr3-ha

Question

In Exercises 16 and $17,$ mark each statement True or False. Justify each answer. 
a. A cube in $\mathbb{R}^{3}$ has exactly five facets.
b. A point $\mathbf{p}$ is an extreme point of a polytope $P$ if and only if $\mathbf{p}$ is a vertex of $P .$ 
c. If $S$ is a nonempty compact convex set and a linear functional attains its maximum at a point $\mathbf{p}$ , then $\mathbf{p}$ is an extreme point of $S$ .
d. $\mathrm{A} 2$ -dimensional polytope always has the same number of vertices and edges.

Chris Trentman · Accepted Answer

forgiven a statement and were asked to market tour false and then to just fire answer in part a. The statement is Cube in. Our three has exactly five facets. The statement is false. A number of facets of a cube in our three is six because in our three, a facet is the same as a square face. Now, in part B, the statement is a point. P is an extreme point of a poly tope if and only if p is a vertex of the politics. This statement is true. This is a direct consequence of the&#x27;re um 14. In the book, the serum says that a point of a poly tope is an extreme point if and only if it&#x27;s a vertex and definitely if it lies in the profile now in part C were given the statement if s is a non empty, compact convex set and a linear functional attains its maximum at a point p then P is an extreme point of s. This is false. So while the maximum is always attained at some extreme point maybe other points that are not extreme points I wish the maximum is attained. In fact, there are examples given in the book. I would say, for example, look, at example three part C. After the&#x27;re, um 16. We see that the linear functional attendance maximum value at to extreme points, but then also at every point in the complex whole of these extreme points and these points in the comics whole are not. Extreme points now in Part D were given the statement a two dimensional Polito always has the same number of Vergis is and edges. This statement is true to see why. Consider Oilers formula so by Oilers formula checking in to be to We get that for a two dimensional polito f zero of call this P and then minus If one of P is equal to one plus negative one to the first, which is zero So it follows F zero p is equal to F one of p. Therefore, the number of Vergis is for this play. Tope is always the same as the number of edges

Gideon Idumah · Answer

so we have some show of force. Question. We&#x27;ll see that the A part is true because this is a symptom of US definition given on pains. 4 57 The be pot is also true. This is you can see Sarah AIDS on page 4 50 seats. The Sea Potts is also true because this is exactly this sister insane. That&#x27;s his car, it&#x27;s your dog returns.

Chris Trentman · Answer

were given statements and were asked to market these statements, tour falls and to justify each answer statement. A Is there are symmetric matrices that are not orthe orginally diagonal Izabal. The statement is false. This is by fear, um, to from this chapter call it the arm to says that a is a symmetric matrix if and only is is or thought Nelly Bagnall Aysel statement B is If the Matrix B is equal to p d. P transposed he transpose equals P in verse and D is a diagnose matrix, then be is a symmetric matrix. This statement is true. See where this statement is true. Recognize that the requirement that B equals P d. P transposed repeat transpose equals P inverse India&#x27;s a diagonal matrix. This means that be is or Thorgan Aly Diagonal Izabal and again by fear, um, to from this chapter we have that be must be symmetric statement see is an orthogonal matrix is orthogonal e diagonal Izabal. This statement is false. This indeed may not be the case Recall that the definition of Martha Graham matrix it&#x27;s matrix who is rowing column vectors or since of worth the normal vectors. So for example. Let&#x27;s take the matrix A to B. Matrix one. Okay, it&#x27;s 2111 one 01 No, 100 01 zero. Yeah, maybe a simpler example. One of her, too. One of a route to and negative one of a route to one of a route to now, just looking at this matrix we have that if we take you one to be the vector of one of a route to one of her too. And you to be the column vector negative 1/2, 1/2. We see that both of these are unit vectors. And that you won dotted with you too, is equal to zero so that they&#x27;re also orthe ogle to each other. So that&#x27;s you one. You too is an Ortho normal set. Likewise, if the one is the column Vector one of a route to one of a route to in v two. Sorry, the row vector, whatever to negative One of her two and V two is the row vector one of the route to one of a route to then we have that V one dotted with V two is also equal to zero and that view and envy tour. Both unit vectors do that V one and V two Foreman Ortho Normal said as well. And so we have that all is that a is an Ortho normal or distant orthogonal matrix. But clearly they transpose is equal to whatever route to one of two on the diagonal. And then we swap negative one of a route to with one of them too. This is not equal to a so that a is not symmetric. Therefore, again, by theory, too, we have that a is not or Thorgan Aly Diagonal Isa ble statement D says the dimension of an Eigen space of a symmetric matrix is sometimes less than multiplicity of the corresponding night in value. This statement is false to see why for to the spectral fear, um, for symmetric matrices, which is three earned three in this chapter and we

Chris Trentman · Answer

were given statements and were asked the label them Sure false and to explain in part, a statement. Is that an n by N matrix that is orthogonal e Dagnall Izabal must be symmetric. This statement is true. This follows from here Um too from this chapter which says that and then by n Matrix A is or Thorgan aly diagonal Izabal if and only if a is a symmetric matrix part B The statement is if a transpose equals a in defectors U and V satisfy a U equals three you and a V equals four d then you dotted with V equals zero. This statement is true noticed that since a transpose equals a we have that is symmetric therefore by theorem one from this chapter it follows that Eigen vectors from distinct wagon spaces are orthogonal to each other. And we have that since a U is equal to three. You and eighties equal for V. We have that you and the are Eigen vectors of a associated with the Eigen vectors for Eigen values three and four, respectively. These are distinct. I&#x27;d in values so by fear and one it follows that you one is orthogonal to or use or Foggin all to be. And so it follows that the doctor product of U and V is going to be zero in part. C Statement is an n by n. Symmetric matrix has end distinct really Eigen values. This is false Indeed. An example in this section included a symmetric matrix a which did not have and distinct rial agon values. This was example three. From this chapter, a true statement would be that the end by in symmetric matrix has and realizing values accounting for Multiplicity. Statement. D says that for a non zero vector V in RN the matrix be transposed. It&#x27;s called a projection matrix. Need to make sure this is so. There&#x27;s actually attend. Um, this should be matrix vv transfers Kidding. And this is false. See why I recall the definition of a projection matrix we have. The projection matrix is a vector. Well, the matrix such that the projection matrix times a vector and are in, is the orthogonal projection of that victor onto the subspace spanned by you. And the reason this isn&#x27;t true is because we need you. Could be Ortho normal Eigen vector. Simply not the that This would be true if the Waas and a member, uh, a set of or so normal I in vectors of the Matrix A Not just any matrix, but a symmetric matrix, but this is not true.

Gideon Idumah · Answer

cinemas. All we have that the A part is force because I find off s equal toe s and they can see uh, the park of below the definition on page for 59 number B parts. This is true. Um, scenes, The fine all off s always contents s. We know these. So if, as contains, it&#x27;s fine all it&#x27;s fine. All So we have a friend all always convinced ISS. So if s contents are fine, all this would imply that s with after Beatie quote So they find all off a fine of s on DDE bites urine Sue serum sue because s a course to a fine of s. This implies that s is our friend Number C is also true. You can see the Parker off below the definition in page for 42. Question is force This statements the statement is only true is all the true when floods he&#x27;s in our three on then is true because floods is a trust leads off subspace. So this means that so if we translate it back if we translates, it&#x27;s Bach to the origin. It becomes a subspace ce good

James Chok · Answer

okay, In this question, we will answer with a Siri&#x27;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&#x27;m not empty district sets such that a is compact and being close. Then there exists a high plane that&#x27;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&#x27;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&#x27;s a hyper thing which week, which, with we can separate these two, there&#x27;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&#x27;t sit right them.

Problem

Let $\mathbf{v}$ be an element of the convex set …

Problem 17 Medium Difficulty

Answer

a) False
b) true
c) False
d) true

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Calculus 3 Educators

Heather Z.

Kayleah T.

Samuel H.

Michael J.

Calculus 3 Courses

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 8

Video Transcript

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Michael J.

Vectors Intro

Vector Basics - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

a) Falseb) truec) Falsed) true

Linear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Michael J.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Michael J.

a) False
b) true
c) False
d) true

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications