3. Let {e1, e2, e3} be the standard basis of C^3, and {v1,...

3. Let \{e_1, e_2, e_3\} be the standard basis of $C^3$, and \{v_1, v_2, v_3\} be the basis $v_1 = \begin{pmatrix} 0 \ 1 \ i \end{pmatrix}$, $v_2 = \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}$, $v_3 = \begin{pmatrix} 1 \ 0 \ i \end{pmatrix}$. Determine the transition matrix $P$ from \{e_1, e_2, e_3\} to \{v_1, v_2, v_3\}, and hence determine the matrix with respect to \{v_1, v_2, v_3\} of the linear transformation whose matrix with respect to the standard basis is $\begin{pmatrix} 2 & 1 & -1 \ 0 & 3 & i \ i & 0 & 2 \end{pmatrix}$.

Transcript

00:01 Hello there.

00:01 Okay, so in this exercise we have two bases, the basis b defined it here by these two vectors to 1 minus 3, 4, and the basis s which corresponds to the standard basis in r2.

00:16 These two are bases in r2 of course, and the first thing that we need to do is to find this transition matrix from the basis b to the basis s.

00:26 This can be done easily because basically the new basis will correspond to the standard basis so in that case this transition matrix from b to s it's just a matrix that the columns are just the vectors of the basis b in this case so to one minus three four the reason behind this is that the the standard procedure to obtain this transition matrix is that you need to construct an extended matrix, where first you're going to put the new basis, and in the extended part you're going to put the old basis.

01:17 Once you have constructed this extended matrix, you need to perform some row operations in order to reduce it to the action form, so that in the left part you're going to have the identity matrix, and in the right part, you will obtain the transition matrix from the old basis to the new one.

01:39 So as you can observe in this case, the new basis corresponds to the standard basis, and this is already the identity matrix on the left part.

01:50 And on the right part which corresponds to the old basis, we're going to put the vectors from the basis b.

01:56 But it's just what we have done here.

01:59 So whenever you have this kind of transition matrix, from some arbitrary basis to the standard one, then the only thing that you need to do is put the vectors as the columns of the matrix.

02:16 The next thing that we need to do is to find the transition matrix from the standard basis to the basis b.

02:28 So this is kind of different and in this case we need to do the whole procedure.

02:33 So first we're going to put the standard.

02:36 Under the basis b so that means here putting the vector to 1 minus 3 4 and here on the right we're going to put the standard basis so that means the identity matrix okay so then we need to perform some role operations to this and that means well first we need to put number 1 in this first entry of the matrix and that means the divided the first row by 2.

03:13 And in that case we obtain here 1, minus 3 halves, 1 half, 0 and the other row will be the same.

03:33 Okay, the next step will be putting a 0 below the first 1.

03:38 So to do that we just need to assign to the second row this subtraction of the second row with the first row.

03:46 And we obtain here 1 minus 2 halves 1 half 0 and here 0 11 1⁄ and 1.

04:05 Then we need to transform this 11 half to 1 so that means that we need to multiply the second row by 2 divided 11 so that we obtain here 1.

04:20 Minus three halves one half and zero in this part we will have zero one minus one over eleven and two over eleven finally we need to eliminate where we need to put a zero in the position of the matrix and that means to assign to the first row the first row plus three halves times the second row and then finally here on the left part of this extended matrix we obtain the identity matrix and in this right part we're going to have the transition matrix that we need that is 4 divided 11 3 divided 11 minus 1 divided 11 and 2 over 11 and of course we can rewrite this in a better way so the matrix from the basis s to the basis b will be 1 over 11 times the matrix 4, 3, minus 1 and 2.

05:40 Okay, great.

05:44 The next step will be showing that these two transition matrix p from s to b and the transition matrix from the basis b to s and the transition matrix from the basis b to s, are inverses of one another.

06:11 And not that she did that, the only thing that we need to do is multiply the matrices and show that they are the identity.

06:19 So what we need to do is multiply this matrix with the other matrix that we obtain it at the beginning.

06:30 Okay, so this first matrix, so this first matrix is the one that we just computed in the previous part, so it's 1 over 11, 4, 3, minus 1, 2, and the other matrix is the one that we obtained at the beginning, that is just putting the vectors of the basis b as columns of the matrix.

07:06 So to 1 minus 3, 4.

07:09 And then we just need to multiply them.

07:13 Okay, so we have 1 over 11 of the command factor, and then we need to multiply this...

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