Suppose T : ?³ ? ?³ is a linear transformation such that...

Name: Suppose T : ?³ ? ?³ is a linear transformation such that T([1; 0; -1]) = [2; 1; -4] T([3; 2; 0]) = [5; 14; 3] T([0; -2; 0]) = [4; -8; 0] Find the matrix A corresponding to the transformation T.
Uploaded: 2022-11-25T12:50:30-08:00
Duration: 6 min 31 s
Channel: Diogo Caetano
Description: Suppose T : ?³ ? ?³ is a linear transformation such that T([1; 0; -1]) = [2; 1; -4] T([3; 2; 0]) = [5; 14; 3] T([0; -2; 0]) = [4; -8; 0] Find the matrix A corresponding to the transformation T.

Suppose T : ?³ ? ?³ is a linear transformation such that T([1; 0; -1]) = [2; 1; -4] T([3; 2; 0]) = [5; 14; 3] T([0; -2; 0]) = [4; -8; 0] Find the matrix A corresponding to the transformation T.

Transcript

00:01 Okay, so in this problem, we're given how t acts on three vectors, 1 -0 -1, 3 -2 -0 -2 -0 -2 -0, and we want to find the matrix corresponding to this linear transformation t.

00:13 Just as a recap, the matrix will be a matrix, you know, 3 by 3, and we're going to have three columns.

00:20 And just to recall, the first column will be t at 1 -0 in the usual basis, second column will be t at 0 -1 -0, and last column will be t.

00:30 At 0 -01.

00:31 So essentially we'll have to write each of these vectors as a linear combination of the ones for which we have information and then make the calculation, right? so let me just note.

00:41 So step one, write the basis in terms of the ones that we are given.

00:45 So we are given one zero minus one, one, we're given three to zero, zero, and zero minus two, zero.

00:59 So let's start with the easy right from the last one, we already know.

01:03 So 010 is simply 0 times the first one, 1 0 minus 1, plus 0 times the second one, plus minus 1, plus minus 1 half the last one, right? 0 minus 2, 0.

01:23 So this one is easy.

01:24 Let's look at now, let's say 0 -01.

01:32 Ok.

01:34 So 0 -0 -1, obviously we're going to need to use the first vector.

01:37 Because the only one that has a last component here.

01:40 So the first vector will of course have to be minus one time, the first vector, and then we need to make up for the rest.

01:49 So for the first vector, i need minus one time.

01:51 So i'm going to have a minus one in here.

01:53 So i'm going to need to use this one to compensate for the minus one with a plus one.

01:57 So i'm going to need plus a third of the second one.

02:02 So let's see.

02:02 From here, i will have on my first entry, i'm going to have a minus one, plus one from here.

02:09 So they cancel out to zero already.

02:10 That's good.

02:11 Second entry, i'm going to have a two -thirds in here.

02:15 So i'm going to need to make up with that one.

02:18 So i have two -thirds.

02:19 I need to add one -third of the last one.

02:24 Right? let's just check.

02:25 First entry is going to be minus one plus one.

02:28 We're good.

02:29 Second entry is going to be zero from here.

02:32 Two -thirds, minus two -thirds.

02:34 Zero.

02:35 We're good.

02:35 The last entry is a minus one.

02:39 And finally, for 1 -0 -0, let's see what we need.

02:45 We're going to combine, i suppose, ideally we would like to just use the first one, but then the first one has a z component.

02:54 So the first one will need to be a zero, right? because then there's no way for us to cancel up the z component.

02:59 So it's going to be to be zero times the first one, so we don't get a z component...

Question

Suppose T : ?³ ? ?³ is a linear transformation such that T([1; 0; -1]) = [2; 1; -4] T([3; 2; 0]) = [5; 14; 3] T([0; -2; 0]) = [4; -8; 0] Find the matrix A corresponding to the transformation T.

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