LESSON 15: CAMERAS (LINEAR TRANSFORMATIONS IN TWO DIMENSIONS)
This lesson: we’ll learn what linear transformations are, and how they can be represented by tables of numbers called matrices.
Do now: 1. Write “student response” on your homework marking
2. Write a heading “Linear transformations and matrices” on a new page in your book (but you’ll be working on the whiteboard in the first bit of this lessons)
3. On your whiteboard, write down what transformations you remember from Year 9. Rotations, and what else?
What do they leave the same?
What other transformations can you do in two dimensions which map every triangle into another triangle (possibly different in size, shape, or position)?
We are going to study these transformations. It turns out that we get a neater theory if we omit translations and include only those transformations which take (0,0) to (0,0).
What’s going on here? We are thinking of transformations as “cameras” which map an original into an image without bothering for now about where the image is, so we might as well assume the image keeps (0,0) where it is.
These operations which map triangles into triangles are called linear transformations.
They map lines into lines. Also, we can translate them into algebra in the same sort of way that we translate geometry into algebra with cartesian coordinates. And the equations describing them are linear equations, with just stuff like ax+by in them, no squares or cubes or anything like that.
Look at linear transformations using Gimp. On your whiteboard, draw any shape you like, then what it would look like:
Now we’re going define multiplying two linear transformations A and B by:
AB = doing B, then A
When we’re multiplying numbers, 1 is special, because 1×x = x whatever x is.
What linear transformation is special in the same way for multiplying linear transformations?
When you multiply together two linear transformations, does the order of multiplication make a difference? Does AB=BA?
We can write every linear transformation as:
(x, y) ↦ (ax+by, cx+dy)
or as a table (matrix)
That matrix transforms the triangle with corners at (0,0), (1,0), and (0,1), into one with corners (0,0), (a,c), (b,d). Matrices are just tables of numbers, but structured so that we can add and multiply whole tables. They were invented by the British mathematicians Arthur Cayley and James Joseph Sylvester a bit over 150 years ago.
We’ll mostly look at tables (matrices) with two rows and two columns (called 2×2 matrices). You can also have 3×2 or 4×7 or whatever, or 25-billion×25-billion, matrices. What is a 1×1 matrix?
Find out what these matrices do by drawing diagrams
The transformation defined in numbers by
takes (1,0) to (a,c) and (0,1) to (b,d)
Background: why “linear transformations” in the sense that lines ↦ lines are also “linear” in the sense that the equations describing them are linear (have just x and y in them, no squares, cubes, square roots, anything like that).
Why all linear transformations can be built up from scaling, shearing, and rotation
Let A be the point (a,c) and B be the point (b,d), then the diagram below shows what the transformation does, and how it can be built up from scaling, shearing, and rotation. We have to include reflection, too, if the triangle gets “flipped” (OA ends up on the anticlockwise side of OB).
We can think of reflection as scaling by a negative factor. We need rotation, too, if both axes are turned in the same direction, and shearing, too, if one axis is turned and the other isn’t.
There is an obvious way to add linear transformations:
if T1 is image x = a1 old x + b1 old y, image y = c1 old x + d1 old y
and T2 is image x = a2 old x + b2 old y, image y = c2 old x + d2 old y
then T1+T2 is image x = (a1+a2) old x + (b1+b2) old y, image y = (c1+c2) old x + (d1+d2) old y
LESSON 16: MULTIPLYING MATRICES
Click here to download pdf.
Possible extra activity
A way of seeing what it means to multiply a row times a column, and a different use of matrices.
I’m a greedy, unhealthy slob. I eat only Big Macs and large fries.
My shopping list is a row (3, 2), meaning I’ll buy 3 Big Macs and 2 large fries;
The shop’s price list is a column (2.59, 1.39) meaning a Big Mac costs £2.59 and large fries £1.39.
For the bill I multiply the shopping-list row and the price-list column:
first number in row × first number in column
+ second number in row × second number in column
What does that come to?
What would the sum for the bill be if Big Macs went up to £3 and large fries down to £1?
What would it be if my shopping-list became (4,1), meaning 4 Big Macs and one large fries, and the price-list became (3,1), meaning £3 for a Big Mac and £1 for large fries?
My friend is as much of an unhealthy slob, but less greedy, and wants 2 burgers and 1 fries. We also have the choice of Burger King – burgers £3.79, fries £1.89.
Now I have a 2×2 matrix with two shopping-lists
And a 2×2 matrix with two price-lists.
When I multiply shopping-list matrix × price-list matrix I get a bills matrix
|Bill for me at McD||Bill for me at BK|
|Bill for friend at McD||Bill for friend at BK|
Work out the bills matrix.
Practise matrix multiplying with Exercise 4C, page 81, Q.1-8, and practise matrix addition with Ex 4A, p.75, Q.3, and Ex 4B, p.77, Q. 1-4. (You may want to look at p.76 to help with Ex 4B, and p.74 to help with Ex 4A)
LESSON 17: DETERMINANTS
What does a negative determinant mean? Meaning images of negative area? Huh? What does the “negative” area mean?
Using determinants to help us identify the geometric meaning of a matrix.
Det = 1 => (in Edexcel, though not always in real life) rotation
Det = -1 => reflection
Det = something else => (probably, in Edexcel) enlargement
LESSON 18: HOW TO REVERSE LINEAR TRANSFORMATIONS, AND WHEN WE CAN’T DO IT
Working on the whiteboard in pairs, find:
1. An example of a transformation T you can reverse (meaning you can find another transformation which always takes you from images produced by T back to the originals). You can describe the transformation in words (easier) or in table form if you like.
2. An example of a transformation you can’t reverse.
3. An example of a transformation you can’t reverse but which isn’t the zero transformation
So: when we are doing ordinary arithmetic, everything is reversible except multiplying by 0. In the arithmetic of transformations, as in everyday life, quite a few things are irreversible.
If T is reversible, we call the transformation which reverses it the inverse of T or T−1. If T−1 exists, then does T always reverse T−1 (as well as T− reversing T)? In other words, if
T−.T=I (the identity transformation)
does it always follow that
If so, why? If not, give an example where it doesn’t work.
Invert some rotation matrices
Invert some shearing matrices
Invert some scaling matrices
What pattern do you see?
The pattern suggests the rule for inverting matrices
Practise inverting transformations.
LESSON 19: MATHEMATICAL INDUCTION – MATRIX MULTIPLICATION PROBLEMS
Starter activity: If A is a matrix, A2 just means A multiplied by A.
A3 means either A2 multiplied by A, or A multiplied by A2. You’ll get the same answer either way, though one way may be easier to work out than the other.
A4 means either A3 multiplied by A, or A multiplied by A3, or A2 multiplied by A2 (all the same answer).
And so on.
Calculate A2 and A3 for A=
Practise mathematical induction problems with multiplying matrices.
LESSON 20 (if time): SIMULTANEOUS EQUATIONS
There is a little bit in the textbook about using matrices to solve simultaneous equations. There have never been questions in the exam about this, but it’s possible there might be one next year. Do Ex.4J Q.1 after looking at the bit in the book about using matrices to solve simultaneous equations.