Matrix of a linear transformation

Let me describe here a bit of a fancier way to look at transitioning to coordinates in linear algebra. We will use notations commonly used in more abstract math, and draw so-called diagrams, but accompany them with text to make understanding easier.

Definition

An isomorphism of vector spaces and is a bijective linear map .

Intuition: Isomorphic vector spaces are the “same” up to renaming the vectors. For example, and are isomorphic: the isomorphism is produced by the map .

So, what are coordinates of vectors in linear algebra? We are used to the notion of coordinates in where coordinates were individual numbers in the -tuples of numbers forming . For a point we would call the (real) numbers the coordinates of .

Now consider a finite-dimensional vector space , and let . Let be a basis of .

For the abstract vector space we need to make an extra step as abstract vectors do not have any number representation by default: they are just members of the underlying set . Namely, we need to choose a basis , and then the decomposition of a vector in the basis gives us the desired set of numbers uniquely describing the vector.

Important statement

The map assigning to each vector the -tuple of its coordinates is a linear transformation. Moreover, it is an isomorphism of vector spaces.

Indeed, if we take the sum of two vectors, then their coordinates also sum up; the same works with multiplying by a scalar. The map is an isomorphism because each vector is also uniquely defined by its coordinates: in other words, the inverse map is also defined. [Reference needed]

(High-level) statement

A choice of a basis is equivalent to choosing an isomorphism .

If we transition to linear transformations, we would like to interpret the linear transformation using coordinates as well: this would give us a way to interpret transformations just using numbers.

Writing this as a diagram,

The transformation is defined as the map by going clockwise on the diagram (forget coordinates ⇒ perform ⇒ return to coordinates).

How do these maps work for a given vector ? Let us draw the diagram with a bit more words for clarity:

Now let’s try to understand how should the map work. All of these diagrams tell us one important fact: it is a linear transformation between and (as a composition of linear transformations). And all linear transformations between and can be described as multiplying by a matrix [Reference needed].

Examples

Building a matrix of a transformation between and (recall MAT223)

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This is a work in progress.