Practical application for linear transformations

Question

Can anyone help me with a practical application of linear transformations? I am very curious about how can be an application in real life that can be solved with a linear transformation.

Answer 1

A linear transformation is not an application in itself; rather, it is a model. Many real life processes are described or approximated by linear transformations. An understanding of linear transformations can help better understand these processes.

These can range from very simplistic things like scaling a recipe to more complex things like studying the evolution of populations of competing species, or setting up convolutional neural networks.

Answer 2

This question, admittedly, is phrased a little weirdly.

I am very curious about how can be an application in real life that can be solved with a linear transformation.

It is not the case that there is something that can be solved by way of linear maps. You can identify certain "things" as linear maps and use the properties of linear maps to make specific things very easy for yourself.

For example, let $P_3$ be the space of all polynomials of degree 3 or less. Then, we can define a map $T: P_3 \to P_3$ such that:

$$\forall f \in P_3: T(f) = \frac{df}{dx}$$

Clearly, this is linear because it is the case that:

$$\frac{d}{dx}(f+g) = \frac{df}{dx}+\frac{dg}{dx}$$

$$\frac{d}{dx}(\alpha \cdot f) = \alpha \cdot \frac{df}{dx}$$

where $\alpha \in \mathbb{R}$. These are properties that you normally prove within the context of Analysis BUT you can relate them to linear algebra.

It's not so much that linear maps, themselves, are super interesting. They are but recognizing other functions as linear maps often allows you to make simplifications that you, otherwise, wouldn't really be able to make.

Also, these maps tend to have very simple properties which allows us to easily manipulate them. There are, of course, other examples but I think I'll leave it at that for the moment.

Edit

So, you wanted an explicit example of how the concept of a linear map is applied in practice.

One possible way in which this is applied is to geometry. A matrix is nothing but a linear map between vector spaces and we can use certain matrices to study rotations in a plane.

For example, the matrix shown below:

$$\begin{pmatrix} \cos(\phi) & -\sin(\phi)\\ \sin(\phi) & \cos(\phi) \end{pmatrix}$$

is typically used to describe rotations about the origin in a 2D plane. Now, this seems rather weird and doesn't seem like the best sort of application. HOWEVER, we must note that this kind of 'rotation of coordinate systems' happens almost all the time when solving Physics problems.

For example, if you're working with a mass travelling on some slope, it is often better to rotate your coordinate system so that the x-axis is parallel to the slope. In that sense, this sort of matrix proves to be very useful and, certainly working with it involves working with a linear map. I hope this answers your question.

Answer 3

You can think of a linear transformation as a matrix, and vice versa. So, at least in the finite-dimensional case, studying linear transformations overlaps with studying matrices. Thus your question can be partially rephrased with a meaningful though very general focus: “Why are matrices important?”, to which there are numerous answers, which you can look up.

Some simple examples are:

• Rotation matrix. This describes rotation of a vector, which can be useful in many circumstances, like if you want to rotate objects in 2D or 3D.

• Projection matrix. Maybe given a point in $\mathbb{R}^3$ you want to find its “shadow” in the $xy$-plane. Then you can use a projection matrix.

• Permutation matrix. You can swap the order of rows or columns of a matrix. In particular, this is a way to represent one of the elementary row operations, which are fundamental to many things including solving linear systems.

• … and many more.

Matrices also represent systems of linear equations, which has numerous applications.