00:01
Okay, so in this exercise we have we need to find two isomorphisms so the first isomorphism corresponds to we need to find an isomorphism within this the vector space of symmetric and matrices we're going to the node ms of dimension three by three so ms three and the euclidean space of a six dimension now what's going to be the structure of a generic symmetric matrix 3 by 3 well these matrices look like this so first we have the diagonal a b c and then we have elements in the in the upper triangular part of this matrix so say d e f and because it's symmetric we are going to repeat this value so d e f right so these are generic symmetric matrix now how we can map this into r6? in such a way that we have a one -to -one relation and this on two.
01:11
Well basically you can observe that this matrix have as a basis the flowing matrices.
01:20
So 1 -0 -0 -0.
01:25
Then here we have 0 -1 -0 -0 -0.
01:33
0 -01 and then we have the following matrices as basis of this space sorry for that it's going to be 1 and then zeros in the rest then here 0 -0 -1 1 and here 0 -0 1 and the last one is this matrix so basically this 6 -1 so basically this 6 matrices form a basis for this kind of for this space so you can observe that this space is six -dimensional and this is going to correspond to the elements that we're going to map onto our six so the idea is that this map is not that hard to construct this isomorphism will be a b c d and f the map you can observe that both have the same dimension so they are on to and the the second thing is that for any two different matrices or for any two different vectors here we have one for we have a unique relation between them.
03:00
I mean there is we have the following matrix, we have the following vector.
03:06
A, b, c, d, and f is equals to a2, b, 2, c2, d2, d2.
03:18
E2 and f2 this implies actually this equivalent to have that the matrix a b c a b c d e f here is d e and f is equals to the matrix a 2 b 2 c 2 b 2 c 2 b 2 b 2 e2 and f2...