The Hindenburg

Firstly, I have done a leftward version of the original transformation which described by the matrix (1 −1) and 0 1 then I have stretched out space by 2 units in the �� direction and 3 units in the �� direction. This is described by the matrix (2 0). Can you see why? (It might help if you draw it out)0 3

We say that the outcome of applying both of these tranformations is their composition. What you are effectively doing, is multiplying the matrices. To do this, you have to imagine they are functions (which they are) and so function notation applies (reading from right to left). For example:

This means to calculate ��(��) then put it into ��(��). The same applies for matrices:

��1��2

This means do ��2 first and then ��1. With what I said earlier in mind, what do you think is the matrix composition of:

(2 0 0 3)(1 −1 0 1 )

The question is really saying “What is the overall effect of applying a rightward shear (name of the first transformation) followed by a stretching of space?”

It could be put even more simply as “Record the position of �� and �� after both transformations”

Let’s run through this step by step. Firstly, let’s find where �� lands after both transformations:

(1) ������ℎ��������������������������������ℎ����������������������������������������0 ������ℎ��������������������ℎ��������ℎ����������������������������������������������������ℎ������������������������������������������ ������������ℎ����������������ℎ��������ℎ������������������:

1∙(2 0)+0∙(0 3)=(2 0)

������ℎ������������������ℎ����������ℎ������������������ℎ����������������������������������ℎ�������������������������������������������� ��ℎ������������������������ (1) ������������ℎ����������������������������������������:1

−1∙(2 0)+1∙(0 3)=(−2 3 )

��ℎ����,������������������������������������������������ℎ����������������������������ℎ����ℎ������������������ℎ������ℎ������ ����������������������������: (2 −2)0 3

If you don’t understand how we got to this point, go back to make sure you really understand how to find where a vector has moved to after a linear transformation. This is all we have done here: we have looked at where the (already transformed) �� and �� have gone to after the second transformation. Then, we have stored this information in a matrix which perfectly sums up both transformations.

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This matrix which we have just found is exactly the matrix that I put into the computer to generate that second image next to the leftward shear. It is called the composition of both transformations. I thought it would be interesting to include the method which is taught to most students who learn linear algebra, so you can see how elegant this way of doing it really is:

�� ��)(�� ℎ

����+���� ����+��ℎ)=(����+���� ����+��ℎ)

This is a dreadful way to present linear algebra, especially due to the beauty at its core. I hope you find our method of working through linear transformations more interesting than simply crunching a bunch of numbers and looking at the result.

And there we have it. I hope that you were able to follow all that information effectively; it was a lot to take in. most importantly, I hope that you really got a sense for what maths is about: it’s not about the calculation, it really is all about the elegance and the intuition. Maths is more of a creative subject than most will think, I would even argue that it is an art along with music and others. It is a way to tell a story and to gain an understanding of the universe. Maybe maths isn’t your favourite subject, but I hope at least you have seen a different side of it and can look upon it differently. For me, this was truly beautiful, don’t you agree?

- Elliot R (Re)