The Student Room Group

FP1 Matrices linear transformations

Hi

Can someone please explain what a linear transformation is - I am looking at exercise 4D and cannot figure out why they are saying some of the matrices are linear transformations and some aren't - totally confused lol

For example:

This is apparently a linear transformation :

Screen Shot 2013-11-28 at 13.48.15.png

totally foxed - :s-smilie:
Reply 1
If you have an (x,y)(x,y) plane a linear transformation is defined as

ax+by=Xax + by = X
cx+dy=Ycx + dy = Y

where a,b,c,da,b,c,d are constants. This has a natural expression in terms of a matrix equation:

Unparseable latex formula:

[br]( \begin{array}{c,c} a & b \\[br]c & d[br]\end{array} ) \lf( \begin{array}{c}[br]x \\[br]y[br]\end{array} \ri) = \lf( \begin{array}{c}[br]X \\[br]Y[br]\end{array} \ri) [br]

.

So Matrices express linear transformations by definition. I'm a bit confused how a matrix isn't a linear transformation. What are the exact wording? Where do I go to see exercise 4D?
(edited 10 years ago)
Reply 2
It is in the FP1 edexcel book - will try and get a screen shot of it xxx
Reply 3
A linear transformation is one where only linear operations are performed on the old x and y values to get the new ones. In practice, this means you can't multiply any two variables; x cannot become xy, y cannot become x^2.

they also have the other property that if operated on the origin (0,0), it will not move.

I was confused by 4D as well. if it helps, think of it this way; for a transformation to be linear, you can only add x and y terms and you can only multiply by constants. so any (x+n) where n is a constant like 1, 3, 5 etc. is not linear because if operated on the origin, it would move the origin.

heads up for when you do matrix transformations; think about the matrix being operated on the identity matrix, and think of the identity matrix as two coordinates of (1,0) and (0,1). if this last bit isn't clear, don't worry just yet, just make sure you understand what a linear transformation is first.
Reply 4
Original post by SubGK
A linear transformation is one where only linear operations are performed on the old x and y values to get the new ones. In practice, this means you can't multiply any two variables; x cannot become xy, y cannot become x^2.

they also have the other property that if operated on the origin (0,0), it will not move.

I was confused by 4D as well. if it helps, think of it this way; for a transformation to be linear, you can only add x and y terms and you can only multiply by constants. so any (x+n) where n is a constant like 1, 3, 5 etc. is not linear because if operated on the origin, it would move the origin.

heads up for when you do matrix transformations; think about the matrix being operated on the identity matrix, and think of the identity matrix as two coordinates of (1,0) and (0,1). if this last bit isn't clear, don't worry just yet, just make sure you understand what a linear transformation is first.


thanks - that makes a lot more sense than the book ! :smile:

Quick Reply

Latest