Can you visualize linear transformation vs. matrix?

We studied matrices in grade 12 and most of us mastered the basic calculation methods quite easily. We moved on to higher grades and used matrices wherever required. However, many of us missed out on getting the visual understanding of matrices.

With so much happening in the field of data, we are surely going to bump into Matrices during our professional journey. In today’s world, where so many software tools are available, implementation is not an issue. However, I feel that implementation becomes more exciting if we are able to visualize it!

Matrices represent linear functions between spaces, called vector spaces. The two fundamental facts about matrices are that every matrix represents some linear function, and every linear function is represented by a matrix.

Firstly, let’s get comfortable with 2-dimensional vector space. Once, that is crystal clear, we can extend the same understanding to higher dimensions.

M :   R2   ——>     R2 This matrix transformation M is moving [ x  y ] to [x y]

Let’s look at a few simple transformations. To understand how does a matrix transform any point in xy plane (2 – dimensional vector space ) , it is sensible to start with the basic vectors of R2.  Let’s define our 2 basic vectors:

e1 = [ 1   0 ]

e2 = [ 0   1 ]

• Stretching Transformation S(e1) = e1

S(e2) = 2e2

If we plot it on the graph, we can see that S is stretching the points along y axis. Yes, it will stretch the points along x axis.

• Reflection Transformation R(e1) = – e1

R(e2) = e2

We can see that R is reflecting the points along x-axis. Yes, it will reflect the points along y axis.

•  Mixed Transformation

We can combine the above 2 transformations to craft a new one, SR, which will stretch & reflect the points. Try to calculate SR(e1) & SR(e2) & plot the points on a graph. You can see stretching & reflection in action.

If you can visualize the above 3 examples, you are all set with your basics. Now, all you need is to build up towers using these basic building blocks. Please find a few visual explanations  of 2×2 matrices as linear transformations. This will make the understanding much more clearer. Enjoy exploring linear transformations vs. Matrices !!