Linear Algebra for Computer Science

Question

What type of linear map is the matrix?

Answer 1

A map is defined as a way to correlate certain objects to every element in a set. For example, a map f from set A to set B is a function such that for every element in a set A, there is a unique object f(A) in a set B.

A linear map is also known as a linear transformation, and is defined as a map that respects addition and multiplication. That's it.

So in the sample example above, let's define a linear map f for two sets A and B. In order to define them as a m x n matrix (m rows and n columns), however, because we added these operations (+ and *) we have to include two conditions for those two sets: they must be vector spaces, and they must have a basis. What is a vector space? It's a set of objects, let's call it V, called vectors, which follow those rules of addition as well as scalar multiplication (multiplication by a non-vector constant). A vector basis B is a set of vectors in that set V which can be used to re-create any of the vectors in V using scalar multiplication and addition. Oh and finally, in order for the linear map to work, A and B must also have a number of elements that isn't infinite.

Mathematically we write this as for vector spaces A and B, the linear map f is: f(A+B) = f(A) + f(B)

and for multiplication, we talk about scalar multiplication, so f(c * B) = c * f(B) or f(c * a) = c* f(A).

So to your question: if you have two vector spaces A and B with bases, any linear map from A to B can be represented as a m x n matrix. So matrices are a useful way of representing examples of linear maps, and working with them mathematically.