Transformations of Matrices (FP1)

I find these really hard for some reason. I don't understand what a linear transformation is, nor do I understand how to work out the transformation the matrix has undergone. Please can somebody help?

Reply 1

Locke4

If you need to work out what transformation a Matrix represents, the best way to do it is to use the Unit Matrix (Not sure if you've encoutnered this before, its a square with vertices such that A: (1,0), B: (1,1), C: (0,1) that forms a square. You then just imagine that the point A is the first column in a 2x2 matrix, and the point C is the next column, forming this matrix:

(1 0)
(0 1)

Then you just "Multiply" this matrix by the matrix of your transformation (if it is a 2x2, not sure what exam board you're on), but you don't need to actually do this calculation as it is just the Identity matrix. You then still have the first column being point A, and the next column being point C, and you do a little sketch to see how this has changed. For example;

(0 1)
(1 0)

This is the first column being point A, which is now (0,1), and the second column being point C, which is now (1,0). Drawing a pair of sketches shows that it is a reflection in the line y=x. Hope that helps! :smile:

Reply 2

alygirl

OP

2

Original post by Locke4

If you need to work out what transformation a Matrix represents, the best way to do it is to use the Unit Matrix (Not sure if you've encoutnered this before, its a square with vertices such that A: (1,0), B: (1,1), C: (0,1) that forms a square. You then just imagine that the point A is the first column in a 2x2 matrix, and the point C is the next column, forming this matrix:

(1 0)
(0 1)

Then you just "Multiply" this matrix by the matrix of your transformation (if it is a 2x2, not sure what exam board you're on), but you don't need to actually do this calculation as it is just the Identity matrix. You then still have the first column being point A, and the next column being point C, and you do a little sketch to see how this has changed. For example;

(0 1)
(1 0)

This is the first column being point A, which is now (0,1), and the second column being point C, which is now (1,0). Drawing a pair of sketches shows that it is a reflection in the line y=x. Hope that helps! :smile:

I have learnt the different rules and so I understand to find the transformation you multiply by these, but how do you determine which transformation has taken place?

Reply 3

Dingo749

9

You need to understand 'special angles, the exact values of sin, cos and tan fot 30, 45, 60 and 90 degrees.

You can work these out by taking a square with sides of 1, cut it diagonally and you have a triangle with two sides 1, the hypotenuse is (sqrt)2.

The 30 and 60 degrees can be worked out by taking a triangle that is equilateral with sides 2. Cut it straight down the middle so you have sides of 1, 2 and (sqrt)3. This is how you remember all the numbers.

From here look at a matrix you, have say its;

|sqrt3/2 -1/2|
|1/2 sqrt3/2|

Now top left and bottom right represent cos and the other two represent sin (This is in your formula booklet) The first thing you do is look for the change of sign, if the sign which is different is on a corner which represents sin it is a rotation, if the change in sign is on a corner that represents cos then its a reflection

This change in sign is top left so its a rotation, so we look at it and say well sqrt3/2 is cos30, check that with sin and yes sin30 is 1/2. So we know its a rotation of 30 degrees, we just need to decide CW or ACW, if the opposite sign is on the right it is ACW, if on the left it is CW.

Therefore this matrix represenys an Anticlockwise Rotation of 30 degrees about the Origin.

For a reflection whatever angle you get, halve it and says its a reflection in the line y=(tantheta) x :smile:

Hope this all helps :smile:

Transformations of Matrices (FP1)

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