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Question

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}ight\}$ be a basis for a vector space $V .$ Explain why the $\mathcal{B}$ -coordinate vectors of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}$ are the columns $\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}$ of the $n 	imes n$ identity matrix.

Doruk Isik · Accepted Answer

in this problem, we&#x27;re given a set of bases and we know the vectors that formed this place is so you know to be and where has to explain why actors are, uh this spaces are the collectors off the on by an identity make. So if this bees are the basis in vector space, be damn You know, that director can be written as a linear combination off these basis vectors. And this it&#x27;s true before device inspector itself. So we can like be one as one time one plus zero b two plus dr 0.0 b m. So don&#x27;t be too would be zero times be one plus one time to be too Plus zero times bien I&#x27;m gonna be an would be zero b one plus zero view too So old zeros up to be end up Enter Amanda and the answer would be so from this, we can find the coordinated vector for anyone to be one old zeros. And from the second equation, be fine reporter factor for me to to be 010 hold zeros and from 30 question we find the court inspector for intern and director to be 000 old zeros. Why intern? So last term is one says you get sick. Or did that reporter? First time is the first column off the identity matrix for a second when it is the second Poland A or B M for it is defense co

Runpeng Li · Answer

All right. So in this problem, work even eight times you equals zero and the times be it was zero. Now, first we need to figure out eight times. You plus B. Now, this is quite simple. We just We know that a matrix matrix multiplication is distributed so unjust that apply the law, which is eight times you plus 800 now. It had serious 08 times v zero. Also, it is zero now. Secondly, we need Thio. Find out many times. See Time&#x27;s Hugh plus the Times V Now C and D are skaters. So again we apply the, um it&#x27;s reputed ity. So that&#x27;s 80 times. See, do you plus eight times Dean. Yeah. Now, since Cmdr scale er&#x27;s, we can take the skater impromptu off the expression so that it&#x27;s C times. Hey, you last e kinds a V. Now, right now, I can apply this associative ity off matrix multiplication so that it&#x27;s C times zero us d times zero. So that is zero and

Ernest Castorena · Answer

I get for this question? The give us the vector space of three by three square matrices with real entries and they want us to find the basis for this upset. Where I wrote and columns some are zero Thank you. So what might they look like? How can we describe them? But if this Waas be and this was see in this entry here automatically has to be If we add them together as a rope negative bees of tracks e I think it. And if this Waas let&#x27;s a a then when we some of this road has to equal to zero this has to be negative a plus B plus c. And if we saw him going to the regular, this was let&#x27;s say D then this has to be blind to the right negative ain&#x27;t minus t and similarly down here everything&#x27;s determined. We have Ah, we have stsnegative. See, That&#x27;s right. That is a plus D minus e and this one has to satisfy two equations. It has to be the when you add negative A plus B plus c onda a plus the man a scene and this new unknown this new entry we got zero cause that things cancel out for a bit. These days will come. So he sees will cancel out. And I&#x27;ll just get a B plus, dear Meaning. So this has to be a minus. B minus D. Some and its be minus stink, and we can see that. Okay. Make sure these Oh, some zero going this week in this way. Yep. Going this way. Going this way yet? We just saw that going down this way. Yes, they do. Going down this way? Yes, they dio. So every columns on Mr Zero Every Rose. I&#x27;m Mr Zero. They looked like this. And guess what? They&#x27;re completely described in terms of a, B, C and D. So we can guess that. Just takes four injuries to describe any, um maitresse Ian the subspace. That&#x27;s so let&#x27;s see. We&#x27;ll use our tricks of reform. This will be a negative. Bean, B zero zero uh zero zero visit. Be here. Is it be here? There&#x27;s no be here. So that&#x27;s one injury about all the A&#x27;s 0001 You&#x27;re a negative one. I m in a and negative. A 00 Does that look cracked? There&#x27;s an area here. Okay to my letters, where all our sees zero c 000 sees your negative scene and one more. And we&#x27;ll have broken down our matrix over here into four other matrices over here. 000 01 negative one zero negative one and one apps. I mean, the I&#x27;ve been getting ahead of myself. Why is that? We can pull out to be here, and they all turned the ones we can pull out the A here. They all turned the ones we can pull out of the sea here and be here. And we&#x27;d see the arbitrary matrix is actually a linear combination of these matrices of 00010 Negative one negative. 101 It&#x27;s a linear combination. Uh, negative. 11001 negative. 1000 Now, then, your combination of negative 10100010 negative one. And lastly, this 100000 one negative one negative 11 And this basis appears just from my choices for how I determined an arbitrary matrix in this subspace us. A different choice would lead to a different basis. It&#x27;s fine as long as they were independent on this planet. Please do. We saw that. Okay, so we know it&#x27;s dimension, for this is the basis and they want us to extend this to a basis for V. What can we do that? No, sir. This is nine dimensions. What we can do using our there. So far, we know that this is definitely a basis for a nine dimensional elector space. If I keep repeating like this, you can complete the induction and see what the last interest it would be. Well, there&#x27;s gonna be nine elements yet you can verify which ones are not a linear combination of these and add them into the collection. And then that way you could extend this to a basis with nine out.

Chris Trentman · Answer

you&#x27;re given a special kind of matrix and were asked to prove statements about this matrix were given that B is an n by n symmetric matrix such that B squared is equal to be. And we&#x27;re told that this is a projection matrix what this is called or an orthogonal projection matrix in some cases. And we&#x27;re told that if why is some victor and are in will denote my white hat matrix speed times the vector Why So why had is another vector and Z will be why minus y hat which is another vector In part they were asked to show that z is orthogonal toe y hat Do this we have z started with why had is equal to well, we can write move these in terms of why so Z is equal to why minus y hat started with and then why hat is equal to B? Why by definition and this is then equal to why minus b why started with you? Why again and we have by the ad activity of the dot product of this is equal to why started with b y minus b y started with do you? Why so we have. This is equal to why started with B, Why minus and then since B is an n by n symmetric matrix and that this is going to be equal to I had previous exercise Why dotted with B oth be of why or be times B times by This is equal to why that it would be why minus why dotted with he&#x27;s squared. Why recall that B squared is equal to be so. This is equal to why started with b y minus why dotted with B y Elizabeth scaler so simply reduces to zero. And so we&#x27;ve shown that Z and white hat are orthogonal in part b were given that w is the column space of B and were asked to show that why is the sum of a vector in W and a vector in W perp or because this w compliment. So prove this. We have that why is some vector and are in And so why? Why Hat started with BU. So we have that bu is gonna be an element w for some Dr you in Oran. It&#x27;s what means to say that it&#x27;s in the column space and this is equal to again since music symmetric matrix This is the same as B Why? Why Hat started with you which is equal to recall that why hat is B y says his be times Why times e times why died with you And we can regret this Since b squared is equal to be as be Why be be why started with you we have this is equal to transpose of e b why you is equal to by the laws of transposition Why transpose the transposed be transposed Why transposed be transposed times you and we have that since he is a symmetric matrix This is equal to why transpose b times be by transposed being times You gonna revise a little bit My proof here so actually left out something important this should be instead of why Why had this is why minus y hat bu which can then be written as b times y minus y hat Just say mus be wind minus the white hat dot you which is unequal to B Y minus. This becomes b squared. Why definition of white hat tied with you and this is then going to be equal to by the activity of the dot product. But first, this is equal to be why minus B Why, since B squared equals B started with you just simply equal to zero dotted with you is your vector, That is which is zero. So you&#x27;ve shown that you is orthogonal to y minus Why hat? Which means that why minus y hat since its orthogonal to every vector in the calls base of B is a member of compliment of the column space of B Therefore, we have that blind minus Why had is equal to some Decter. I&#x27;ll call it the where V is in the complement of W so that why is equal to why hat plus the where we have that why hat is in the calm space of be and V hats in the complement of the columns Basit B. And we have that in particular the is just equal to the vector. Why minus why hat Next? We&#x27;re asked, why does this prove that B y is the orthogonal projection of why onto the column space of B. Well, we have that by the earth ogle decomposition here. Um, this is a unique decomposition and so it follows that. Why hat speak well to projection of why on to w So we have that. B y is the projection of why onto Collins base of B, which is W.

Doruk Isik · Answer

in this problem, we&#x27;re given a basis for our end. And we know the activism for the basis, or we have the one up to be end. So we have been records. So we know that we can write any magic tricks using a coordinate transformation magics and using the cordant vector. So this means that if you were to supply the vector X with the commercial for, um, Courtney and transformation metrics, then we would get the ordinance vector. So not here. The inverse of this pby, um, is actually linear transformation from Arkin to our end. And as we know, they&#x27;re in vectors in our basis. Be So to do this, we need this, uh, pb vector. And this rector will be in and by majors. So if you know that P B is the old meant it make it&#x27;s formed by the basis specters of B one B two, the end soda and the vector. That rule the major is that we&#x27;re looking for will be the inverse of this matrix and that the p p p chew up to be in heroes said this Phoebe made Texas and my head. And this in worst TV is also invited

Doruk Isik · Answer

in this problem, we&#x27;re given a set of vectors. It was represent those this B and we&#x27;re set up. These factors are the news independence and be half in number of factors and barely needle about independent in end dimensional spice. So it be work you former system out of these wreckers, Uh, you would have a number of factors intimations by sword on the cooling system will be an n by n square metres so that we would have given element for vitro since the number of columns is equal number rolls. And since b vectors are the nearly abundant and since you know that number of columns is think of a number of roles so that this given, ah, system given set up back to roll spend space. We can combine this and say that this given system is actually a faces or, uh, the n dimensional space

Problem

Let $S$ be a finite set in a vector space $V$ wit…

Problem 18 Medium Difficulty

Let $\mathcal{B}=\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$ be a basis for a vector space $V .$ Explain why the $\mathcal{B}$ -coordinate vectors of $\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}$ are the columns $\mathbf{e}_{1}, \ldots, \mathbf{e}_{n}$ of the $n \times n$ identity matrix.

Answer

$b_{1_{\mathcal{B}}}=\left[\begin{array}{l}{1} \\ {0} \\ {\vdots} \\ {0}\end{array}\right]=e_{1}$

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

Linear Algebra and Its Applications

Catherine R.

Caleb E.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

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$b_{1_{\mathcal{B}}}=\left[\begin{array}{l}{1} \\ {0} \\ {\vdots} \\ {0}\end{array}\right]=e_{1}$

Linear Algebra and Its Applications

Catherine R.

Caleb E.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

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

Catherine R.

Caleb E.

Kristen K.

Michael J.

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

Linear Algebra and Its Applications