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16. (6 points) Define T .2,R'by Tlv) = Av where A = isomorphism? Prove YOur answer .Is T an...

Question

16. (6 points) Define T .2,R'by Tlv) = Av where A = isomorphism? Prove YOur answer .Is T an

16. (6 points) Define T .2,R'by Tlv) = Av where A = isomorphism? Prove YOur answer . Is T an



Answers

Prove that if $T: V \rightarrow W$ is an isomorphism, then so is $T^{-1}: W \rightarrow V$

Hello there. In this exercise we have the following mapping between the space of polynomial of utmost degree to And the space of matrices Of they mentioned two x 2. They made the mapping is find as follows, we are going to take a polynomial and we're going to put in the diagonal the evolution of the parliament at zero. And the other day I right now we are going to put the values of the point of being evaluated at the one. Okay, so let's see deeper in the structure. How it's going to look this kind of transformation. This mapping. So we know that uh binary point on your P two is given by The following expression. So it's going to be a zero Plus A one X plus eight two X square. That's the generic structure. No, What's going to be P0 in particular? Well, it's going to be a zero on P one. It's going to be equal to a one plus a two plus 80. So that means that after we perform this transformation t we apply this transformation t our matrix In M- 22. It's going to look like a two eight a zero 81 to us 82 plus 80 A one plus 82 plus 80 and 80. So far it looks good. Now the point is that if we want this transformation to be an icy murph is um we need to show that this is This relation is 1- one. So let me consider the following two Polonius. So the first problem you'll will be equal to a zero plus a one X Plus 8 2 Extra Square. I'm going to consider another point on you will be too this point of view will be given again here by easier plus 82 X plus a one X square. Now, you know that this point of view in particular give you as resolved this matrix. Now, what's going to be the result of this point? I'm after applying a transformation? Well, it's going to be the full with matrix A zero, A one 82 plus eight one plus 80. A two plus a one plus 80. And here is not a Tuesday 80. I'm here easier. Okay, so what you can observe what basically these two matrices are the same. And we know that if the relation is 1-1, that means if these two matrices are the same, these two point animals here should be the same. But that is not the point in this case. These 2.0, mules are clearly different between them. So that means that we have to point animals that give you a result the same matrix. This is not a 1-1 relation and therefore is not a nice dwarfism


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