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
00:31seen these applied onto different
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.