Linear Transformations

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The subject of linear algebra is a staple of every undergraduate mathematics degree. To me linear algebra is one of the most beautiful parts of mathematics due to its simplicity while at the same time being extremely useful and powerful. The concept of linearity is ubiquitous throughout most areas of mathematics and also appears naturally in subjects such as quantum mechanics.

In this post I will introduce one of the fundamental concepts of linear algebra and that is the idea of a linear transformation. In linear algebra we study objects called vector spaces and linear transformations are the structure preserving maps between vector spaces. The most common examples of vector spaces include n-dimensional real and complex space but there are many other objects besides these that are also vector spaces. For example, we can consider the set of all polynomials with coefficients from a field as a vector space over the same field.

In mathematics a key idea is the study of structure preserving maps or functions between two spaces. Suppose that we have two vector spaces V and W and a function T whose domain is V and codomain is W. We will say that T is a linear transformation if the following two conditions hold for all vectors v, v' in V and all scalars k

T(v + v') = T(v) + T(v')
T(kv) = kT(v)

These two conditions mean that T preserves addition and scalar multiplication which are the two operations available for vector spaces. The most important type of linear transformation is a matrix, however many types of functions are linear transformations. We will now give a couple of examples of linear transformations.

Our first example of a linear transformation is a function from 3-dimensional real space to 2-dimensional real space defined as follows,
T(x, y, z) = (x, y). We can easily check that T satisfies the two properties given above:

T(x + x', y + y', z + z') = (x + x', y + y') = (x, y) + (x', y')

and

T(kx, ky, kz) = (kx, ky) = k(x, y)

This particular type of linear transformation is called a projection because we project the point (x, y, z) on to its first two coordinates (x, y). As another simple example suppose that V is the vector space of polynomials in one variable. We can define a linear transformation from V to itself by taking T(p(x)) = p'(x) where p(x) is a polynomial and p'(x) is the derivative of the polynomial p(x). We know from basic calculus that the derivative operator is linear and so it follows that T is a linear transformation.

Linear transformations have a slew of nice properties and we can't list them all here but we will mention a couple. The first property is that if we have a basis for our vector space and we have defined a function on the basis elements then we know by linearity the value of the function on any vector in our vector space. Other important properties include the fact that the kernel and range of linear transformations are subspaces of the respective vector spaces of the domain and codomain of the transformation.

If T is a linear transformation between finite dimensional vector spaces V and W then we can choose a basis for V and W. By making a change of basis we can interpret the linear transformation T as a matrix. Thus, by studying the properties of matrices we are also learning about linear transformations between finite dimensional vector spaces. It is this fact that makes matrices such an important part of the study of finite dimensional vector spaces.

Can you think of your own example of a linear transformation? Let me know below.

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References:

http://mathworld.wolfram.com/LinearTransformation.html
https://en.wikipedia.org/wiki/Linear_map

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All the images in this post were created by myself using latex.