Learn AIHL 3.9.1 Matrix transformations (intro and stretches) in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 3.9—Matrix transformations.
Learn AIHL 3.9.1 Matrix transformations (intro and stretches) in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 3.9—Matrix transformations.
The video discusses matrix transformations, specifically focusing on horizontal and vertical stretches. It explains how matrices can be used to transform shapes and points, detailing the process of applying these transformations to a triangle using specific transformation matrices.
The video concludes with a promise of additional videos covering the remaining transformations.
00:00Hi guys, okay, so one
00:02of the really nice applications
00:04of matrices is matrix transformation.
00:06So we can use matrices
00:07to transform shapes and points
00:11lines or whatever into a
00:15different shape or point or
00:17line. And we're going to
00:18apply these transformations, so these
00:20six transformations we're going to
00:22focus on. What you may
00:24have seen these transformations, if
00:27you've seen
00:28studied functions, topic, you'd have
00:34functions. Plus in middle school
00:36you may have looked at
00:38transforming shapes by these transformations.
00:42Anyway, we have translations, rotations,
00:45reflections, horizontal stretch, vertical stretch,
00:48and environments. I'm starting with
00:52the stretches, and we're going
00:53to do, I've called this
00:55video intro and
00:56So we'll do the two
00:58stretches together and I'll do
00:59a video for each of
01:00the other ones. I'm starting
01:01with stretches because it's a
01:02very nice one to show
01:04you. It's very easy to
01:05see how it works. But
01:09these formulae are given in
01:11the form of booklet anyway.
01:13So this example says the
01:15triangular's transformed by the matrix
01:163, 0, 0, 1, find
01:18the coordinates of the new
01:19triangle. So the first thing
01:20to look at is which
01:21of these is it? Well,
01:24it's
01:24It's this guy here. It's
01:25the K001. And this is
01:27a horizontal stretch. A horizontal
01:31stretch or our stretch parallel
01:33to the x -axis with
01:35a scale factor of K.
01:37So the scale factor here
01:38will be three. So what
01:40we do is we get
01:42our three, zero, zero, one,
01:46our transformation matrix. And we
01:48multiply it by the points
01:52of the vertices of the
01:54triangle. So this first point
01:56here is 1 1. This
01:59point here is 2 2
02:01and this point here is
02:043 1. And we multiply
02:06out the matrices like this.
02:10So we do 3 times
02:111 is 3 plus 0
02:14times 1 is 0. So
02:17it's just 3. So this
02:19is 3. Then I do
02:203 times 2 is 6
02:22plus 0 and 3 times
02:253 is 9. Why am
02:34I struggling with 3 times
02:353? Okay, so what you
02:38see because of the nature
02:39of how you multiply out
02:40my edges is you do
02:41the 3 times the 1
02:43and the 0 times this
02:441, 3 times the 2
02:46and the 0 times this
02:472 and the 3 times
02:48There's the three and the
02:49zero times this one. So
02:50what you're essentially doing is
02:52you're multiplying the x coordinates,
02:54which are these one, two,
02:55three, by three. So you're
02:59multiplying the x coordinates by
03:00three, which is stretching the
03:02triangle by three. And then
03:04multiplying the y coordinates by
03:06zero. So the y coordinates,
03:08you're adding nothing. So that
03:10just multiplies the x coordinates
03:11by three. And then when
03:12you do zero times zero,
03:14one times one, because the
03:15one is here,
03:16and the 0 is there.
03:18You end up just getting
03:19the same y coordinate, which
03:21is what you want, to
03:22produce 0 times 2, 0,
03:231 times 2 gives you
03:24my 2, 0 times 3
03:26is 0, 1 times 1
03:28gives me the 1. So
03:30what happens is the x
03:32-coordinates get multiplied by 3
03:34and the y -coordinates stay
03:36the same because of the
03:37nature of multiplying, which it
03:39says. And that's it. So
03:40the new coordinates, it says
03:41find the new coordinates of
03:42the triangles of the new
03:43coordinates are
03:44Or it may phrase this
03:46in a particular way. That's
03:48a, let's see, find a
03:51-b -c - or whatever.
03:53A - would be 3
03:57-1. So this is the
03:59core. This is like a
04:00position vector if you like,
04:02but the coordinate is 3
04:03-1. The coordinate of b
04:06-6, 2, and the coordinate
04:10of c -1.
04:12C dash is like the
04:15image of this point would
04:17be 9 .1. If you
04:23were to draw the triangle,
04:24you'll see that this point
04:26goes to 3 .1. This
04:28point goes to 6 .2,
04:29which would be like here.
04:32And this point goes to
04:329 .1, which would be
04:34over here. So you can
04:37see, let's just do that
04:38again, you can see
04:40this is a horizontal stretch.
04:46We have stretched this triangle
04:47horizontally with a scale factor.
04:51Okay, obviously guys, they can
04:53they can ask these questions
04:54in different ways. Like they
04:56could actually give you the
04:59the image and say what
05:00was the matrix. And when
05:02they do that, you have
05:03to figure out right is
05:03a horizontal stretch. The scale
05:05factor is three formula book
05:07that's here. This is the
05:08one, one, and K is
05:09three. Easiest thought. Okay, second
05:13one, same triangle, but this
05:18time we've two at the
05:19bottom. So it's basically the
05:20same thing guys, it's the
05:21same reason that it's gonna
05:23happen. But the two, the
05:26one is at, in this
05:27position and the two is
05:28in this position. So what
05:29it's gonna do is multiply
05:31the y -coordinates by two.
05:33And I'll show you that
05:34in action here. Now, same
05:36coordinates are gonna
05:36do 1, 1, 2, 2,
05:393, 1. Multiply these. So
05:42the 1 times 1 and
05:45the 0 times 1 is
05:47just going to give me
05:481, 2, and 3. Those
05:51x coordinates stay the same
05:53because I get 1 times
05:541, 1 times 2, 1
05:55times 3 and the 0
05:57just adding 0 does nothing.
06:00And then for the new
06:03y coordinates, I'm going to
06:04get my 0
06:04times 1, or my 0
06:06times 2, or my 0
06:06times 3, or my 0
06:07times 3, or it does
06:07nothing. But the 2 times
06:091 gives me the 2.
06:10The 2 times 2 gives
06:12me the 4. And the
06:132 times 1 here gives
06:16me this 2. So now
06:17the y coordinates, which were
06:191, 2, 1, have been
06:21multiple by 2, which is
06:222, 4, 2, which gives
06:24us a vertical stretch of
06:29scale factor 2. And that
06:31means draw.
06:33Let me draw it. Let's
06:35use this. So now my
06:37one one goes to one
06:38two so that gets vertically
06:40stretched up to. My two
06:42two goes to two four.
06:44Should be like here. And
06:46my three one goes to
06:48three two. So the x
06:51-coordinates are the same, but
06:53the y -coordinates. I like
06:55this. So this is a
06:56vertical stretch. Again, you could
06:59be asked
07:01You could be asked to
07:02find the matrix or you
07:04could be asked to describe
07:05the transformation given by this
07:07matrix when they say to
07:08describe the transformation. You look
07:10at the matrix, you go
07:11to here and say, right,
07:12which one of this is
07:13it, you go, okay, it's
07:14this guy. So it's a
07:16vertical stretch and the scale
07:17factor is two. Okay, that's
07:21the introduction guys. I'll do
07:23four more videos on these.
07:26One for each of the
07:29four
07:29or anyone, so, I'll see
07:30you then.