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Point) Consider the following two ordered bases of R?{(1,-1,-1) , (-1,0,1) , (-2,0,1)}, {(1,-1,1), (-1,0,-1) , (-1,-1,0)}. Find the change of basis matrix from the ...

Question

Point) Consider the following two ordered bases of R?{(1,-1,-1) , (-1,0,1) , (-2,0,1)}, {(1,-1,1), (-1,0,-1) , (-1,-1,0)}. Find the change of basis matrix from the basis B to the basis C.[idjgb. Find the change of basis matrix from the basis € to the basis B.[idJE

point) Consider the following two ordered bases of R? {(1,-1,-1) , (-1,0,1) , (-2,0,1)}, {(1,-1,1), (-1,0,-1) , (-1,-1,0)}. Find the change of basis matrix from the basis B to the basis C. [idjg b. Find the change of basis matrix from the basis € to the basis B. [idJE



Answers

Find the change-of-basis matrix $P_{C \leftarrow B}$ from the given ordered basis $B$ to the given ordered basis $C$ of the vector space $V.$ $$\begin{array}{l}V=\mathbb{R}^{3} ; B=\{(-7,4,4),(4,2,-1),(-7,5,0)\} \\C=\{(1,1,0),(0,1,1),(3,-1,-1)\} \end{array}.$$

I like it for this question. They want me to determine that I mentioned of the North Space Given Matrix A In this case, they've given us one negative 236 What can we do? Well, we can actually just compete the rank of the matrix of thinking, cause we know that the ring of a plus that I mentioned of the North Base is equal to the dimension of the code. I mean, that's the rank Naledi them yet Well, looking at our matrix we actually see of Ah, this is just the first column multiplied by three so we could eliminate actually ended on the Swiss 13 00 And we'll from here it's pretty clear that the rank of the matrix is just one. And while the co domain this is the two by two matrix is they mentioned to the one plus that I mentioned were no space. It's too, so we can subtract one from both sides and we know that the dimension that we want is one

We need to find the change of basis metrics from B to C. So first we need to equates a fair specter of basis. Be the cheese to negative by zero on. Find the victor that has three components C one c two C three that converts the or the basis see to be so Here we have C one supplied by fair Spektr on one plus c two 20 one plus c three 013 So here your cheeks. Why, I said. And then X is a call to see one by, except the first victor see to my way the first x of the second victor and similarly for the third victor. So we have a system of equations. Three equations see one thus to see to close to two negative C one plus C three, because to negative fine. I see one plus C two plus three seats. Three is close to zero. Here, you can just use one of thes a different color. So you have C three close to C one plus C minus five and then replace C three. Come over here, or we can also use because in elimination and documented metrics to solve it. It's ah, just a Z self aversion to solve. So we'll find. See one close to four C to close to 91 c three because to unit one on the first Pictoris for a 1 81 So this was the first factor you need to, right? The other one. Jeez, the second Muktar reaches 305 and same s. We did see one by one and round one plus C two Choose your one C three video 13 So should not leave you right. This isn't questions for the second victor se see Juan Plus to see to close to three, then the for why he writes native see one plus c three because +20 on lastly, see one plus c two plus three c three these because to fight here we can use C three because to see one So it's very simple. You just replace, uh here and then we have system of questions off two equations so it's easy to solve on you'll find C one c two c three to be one one I want So the second victor would be what is wrong now we need to write the third victor on Equated to the other basis. See? So we have h needed to maybe nine. It goes to I see one bye bye. One the good one plus C two by 20 born plus c three 013 And we write this stuff equations Your I see one plus to see to because to age And if you want. Plus, C three is, of course, a negative, too. See one plus C two plus freeze C three It's close Winning nine and yeah, smart before C three Coster Negative too. Plus you wan on, we can replace it here and we will have this itself. A question to a question So four c one c two C three to recall so they have to by on and maybe four. So the fares victor will be native to five and negative four. So these vectors we have weakened three vectors these to about here each are one Colin off the metrics were looking for. So the final, um, answer which is the change of basis metrics from B to c. R. A. T you b two c is a call to on three columns that we had for native want born one on one and then to by negative for and this is the right now.

This question is a little less computational. Lee Heavy. Ah, See the information that we want from the problem? They just want us to find the dimension of the north space of a given matrix. Okay, well, let's see if our matrix is a negative. 933 negative five. This is a one by two matrix. So it should be out putting five dimensional vector. However, we only have one entry to express leg. The possible solution. That means that five subtract one for therefore, um degrees, which are completely ignored by our matrix. So the null space will definitely be for and the reason why that helped. Because we knew the range waas one. And we know that if we had that I mentioned of the rain and the dimension of the hospice, we should get the total dimension of the cold. I mean,

We need to find the general basis metrics from B to C the 1st 1st we need to write the A fierce victor off basis P, which is negative. Five negative three. And then you need to find Victor, be to competency. Want to see to, uh, been supplied by, um the basis see is going to give us this. So I write it down. She won by by six into the first picture off basis. See on seating Bye bye. One a negative one. So here this is X is why and so see one for by X of the first Victor Setem Bye bye. Accept it. Second Victor is going to give us X in the, um, basis B b. So here, right system for equations If you have 61 plus c two physical to negative five on again juicy one minus C two gives us negative three and you just rewrite this one, for example, we say she want is a call to negative three plus C two divided by two and then replaces here and we find C one C two beach are and it won on one. So the first victor, he's We won to see you get one one. Now we need to quit the second victor off. You're that basis be Find another picture for N 28 equals two c one R U C one and C two again she two plus C two one and needed one. So here your rights and other system of equations six. C one lost C to musicals before and six c Juan nine and see two musical Citrus aids. So we say folks here, right Color C to culture to AIDS minus Sorry. Negative to a plus, uh, six C one, then it replaces. Here we define C one and C two as four on negative 20. So the sick and victor for on maybe 20 then the These are the columns off the change of basis metrics. So from B to C, he's 11 four on negative 20. So we should be I know the answer


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