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HW4.6: Problem 6Previous ProblemProblem ListNext Problempoint) Let B be the matrix Cefined by~H 33Find rank(B) by either computing basis for the column space of B o...

Question

HW4.6: Problem 6Previous ProblemProblem ListNext Problempoint) Let B be the matrix Cefined by~H 33Find rank(B) by either computing basis for the column space of B or by computing basis for the rOw space Of B.rank( B)

HW4.6: Problem 6 Previous Problem Problem List Next Problem point) Let B be the matrix Cefined by ~H 3 3 Find rank(B) by either computing basis for the column space of B or by computing basis for the rOw space Of B. rank( B)



Answers

Repeat Problem 9.14 for the matrix $B=\left[\begin{array}{ccc}3 & -1 & 1 \\ 7 & -5 & 1 \\ 6 & -6 & 2\end{array}\right]$ (a) First find the characteristic polynomial $\Delta(t)$ of $B$. We have (b) Find a basis for the eigenspace of each eigenvalue of $B$.

We need to find the change of basis magics from basis. See, to be so the first victor off order basis. See ease. E true to then we need to find the victor that with four components you want C two c three c four. This rain is by by the victors off the order basis B gives us victory to too. So what we do? These were a c one. Why? By first Victor he told on so on. So here three this problem ease easy to solve. We can just do it by looking here. So we need e true to on what we see here except from sea to the rest should be zero to the zero. The first victor politic to be would be geo 100 Then for the second Victor Green. So I write down the second victor off or the basis See you cheese 11 So for one year it's this one. So c four is one arrested zero he said one. He's the 21 So here is the 3rd 1 said Victor to be thirties wanderer ist He's here on four. There were you or to be you to have a look at the fourth mixer he wants to. So it's 1st 1 The 1st 1 is gone. Jiro, is you here? So the change of basis metrics from C to B C to do we called to all these columns. So here we have the first column. 0100 So can you. 00 one. At ease, Geo, you want jail? So this is the change of bait basis metrics that we needed to find.

So for this question, they want us to determine ah, basis for the subspace. Where to buy teammate Tracy's roll entries again? It's gonna be a subspace of it. Span, I This matrix 13 negative. 12 0000 K negative one for 11 on five. Negative six Negative. High one. Okay, Now, at Adela or what? I spent any major see that can be written as a linear combination of these. In other words, the collection of maitresse. Ease where you take some coefficient. Multiply it by this. Choose another coefficient. Multiply by this another another. I want to play them all and add them all together. Well, what happens when you multiply zero by anything? A 00000 vector by anything in this case is here. I make you see Well, you just get the zero major see again. So it's adding absolutely nothing to our span. Well, next thing I don't know if these are independent from looking at them because first off they remain Drissi form. Um but they're completely described by their coefficients so we can describe them as back tears and our Ford Okay, so we can ask if those air linearly independent by singing the solutions to the linear system of them for zero if they're basis. There's only one way to get the zero vector art. Zero Manger's get in terms of traces of scale er's for your vectors. That's right. The mountain there are 13 negative. 12 So I unrolled this matrix Spiro into a same thing for the 2nd 1 Negative. 14 11 I have to keep consistent. Five Negative six negative. 51 Mecca. So maybe there are linearly independent. I'll be able to see that if idea little Gaussian elimination. Ah, we'll have the first spear pivot, subtract three times the pivot row to row two 07 Negative 21. See, I just added Teoh three and we'll immediately CEO thing and ah, let's subtract two of it to row for Okay, well, now, notice these two rows here, it's gonna end up being the case that I'm gonna cheese. This is the pivot, and I'm gonna add it to the other road. So we have one negative. 150 01 negative. 300000 0000 Okay. And Now we can see that only two of these vectors were linearly independent. We have Ah, I can choose the first maitresse e and the second waitressing. Think it. And in this case, what can we do that unravels as one negative 13 to 1 unravel. But it corresponds to these two. Major sees 12 For what? Okay, these are enough to describe our subspace and they're independent. This is a basis for our span. It's only two dimensions because the third can be written as a linear combination of the other two.

To solve this problem could simply ah, compute the inverse off the metrics up in problem 21 But we can also saw directly. So first, yes, your the first victor. The first element off ordered Macy's C, which was two X as and everyone. And then we need to find, uh, Victor, the two companies off C one C two. That that's is when supplied by the victors off. Ah ordered Mississippi gives us everyone. So you're at sea. One supply by so minus four aches plus C two. All right, thanks. So what we have you need to rewrites seven c one minus four c one x plus five c two x If you look, we have one experiment gone Experts who we need to equate the same powers off x So here. Alright. Based on x this force Iran plus five c two bus seven C one. So here this physical to nearly threw a seven to ban. So we have a system of equations. That's why right here four c one plus five c two It's a quote to to on seven See Juan musicals one. So see what he's and be replaced. You want here when we get to see to, he's equal to native to certain. So the Victor Prospector take to the seas being it's going to be on these two. No need to write the second victor, the cheese two plus X with same as we have seen, run by by seven for X less seafood five x. So this system of equations would be same as above. Just the difference is that the coefficients that you're going to yes, so c two on and said I didn't see one here. It's a close to warn Hear music goes to so see one musicals to certain. Yeah, you can get C two, he says. If you didn't realize he's one, too. So these coefficients So this second factor letting to be it's going to be true. Service insurance. These two Richter's that were obtained are the columns off the change of basis metrics from and seem to be so he's a cough. So this is the first victor, you say. So this is the Jeff Base magics that

Okay for this question. They want us to find a basis for a subspace of our three. And by three factors. The span 101 011 and t zero. Thank you. Well, Stephanie gonna be degree three or less, but actually, we see that two of these vectors aren't even linearly independent. This one's just two times that basis. So it's not adding any new information. No new possibilities. Why don't we take a linear combination? Uh, these three effective? Yes. So we can actually just throw this one out and only these to provide new information. It will make a new vector reachable in the spent. So our basis will be 101 and 011 awesome.


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