What is a Linear Map?

The concept of a linear map, or a linear transformation, powerful and beautiful tool and one that is fundamental to (you guessed it!) linear algebra. The idea of a linear map is deeply tied (you could even say isomorphic ;)) to that of matrices and can be understood on many levels. In this article I will try to do my best to introduce the concept of linear maps in a few different ways, each targeted at readers with a different level of understanding, and each focusing on the intuition rather that any algebraic manipulation (although I admit some of that is unavoidable). We will focus on linear maps between to vector spaces, but as I may mention that concept too can be generalized. That being said there are still going to be some perquisite knowledge that I will list below:

• An understanding of functions as a relation between two sets. As well as a familiarity with function notation f: X→ Y. See here.
• An understanding of vectors and vector spaces.

Some of the more ‘advanced’ or abstract understandings may require some additional prerequisites but I will list (link) them as I go! The idea of a linear map is, in some sense, is something that you have likely seen in some form before. I’ll begin by throwing out the definition and then we can interpret it and develop some intuition.

Linear Maps

A linear map is a function between two vector spaces where addition and scalar multiplication are preserved. It is a function that abides by two conditions: additivity and homogeneity. Now what does this mean? If we let function T : V → W, T is linear if:

You make have seen this before perhaps in the context of calculus (the derivative of the sum is the sum of the derivative). Now algebraically this is saying that if you add two vectors u, v ∈ V and then apply the linear transformation it is the same if you had applied the linear transformation to each of them first and then added/scaled the vector.

This algebraic property has a nice geometric interpretation that can strengthen your intuition: namely that a linear transformation can be interpreted as the stretching, squishing or rotating of space such that “grid lines” remain parallel and such that the origin is always mapped to the origin (remains in the same place). Here is a link= a 3B1B/Khan Academy collaboration that I believe shows this beautifully and which I will link here, and here clip below comes from an interactive tool that you should definitely try out here.

A consequence of a map being linear is that the transformation is completely determined by where it takes the basis vectors. You can recognize this as a results of the ‘gridlines’ staying parallel, since if you know where the transformation takes the unit vectors, you can construct the skewed gridlines and thus the entire transformation itself.

The Space of Linear Maps

This next part requires that you have a solid understanding of vector spaces and functions as we will show that the space of all linear maps of the form T: V→ W, denoted L(V, W).

Vector addition in this space the sum of linear maps such that

holds. Likewise, scalar multiplication is the product of a scalar and a linear map T such that

holds. It is clear that adding and scaling linear maps will only create linear maps. It is a good exercise to check that the remaining properties hold, and if you happen to forget what those are check out my article on abstract vector spaces.

So what does knowing that the space of all linear maps is a vector space afford us? It means that linear maps ARE vectors in the appropriate vector space and that we can apply all of the same techniques to linear maps as we could to any vector.