Linear transformations as matrices

Like always, suppose we have a finite dimensional vector space V over a field k. And suppose we have a basis \mathcal B=\{v_1,v_2,\dots,v_n\}. Then if I have any vector v, I can write it as

v=c_1v_1+c_2v_2+\cdots+c_nv_n

where each c_i\in k. In an effort to to be lazy, I could skip out on writing the v_i and just record the c_i (of course, this is entirely dependent on my choice of basis. I am going to write them in a vertical column like so:

v=c_1v_1+c_2v_2+\cdots+c_nv_n=\left(\begin{array}{c}c_1\\ c_2\\ \vdots\\ c_n\end{array}\right)_{\mathcal B}

We put the \mathcal B in the subscript to remind us over what basis we are working. A few basic examples,

v_1=\left(\begin{array}{c}1\\ 0\\ 0\\ \vdots\\ 0\end{array}\right)_{\mathcal B}v_2=\left(\begin{array}{c}0\\ 1\\ 0\\ \vdots\\ 0\end{array}\right)_{\mathcal B}, all the way to v_n=\left(\begin{array}{c}0\\ 0\\ 0\\ \vdots\\ 1\end{array}\right)_{\mathcal B}

But last time we noticed that if we know what a linear transformation does to a basis, we know what it does to the entire space. So if T:V\to W is a linear transformation, I can  write down Tv_1, Tv_2, … Tv_n in this “column notation.”

Let’s say \mathcal C=\{w_1,w_2,\dots,w_m\} is a basis for W. Notice that the dimensions of V and W need not be the same. Since Tv_i\in W, we can write Tv_i=a_{1,i}w_1+a_{2,i}w_2+\cdots+a_{m,i}w_m. In the column notation, we have:

Tv_i=\left(\begin{array}{c}a_{1,i}\\ a_{2,i}\\ \vdots\\ a_{m,i}\end{array}\right)_{\mathcal C}.

But we have one of these for each i=1,2,\dots n. There’s no sense in writing so many parentheses and subscripted \mathcal Cs. Let’s just concatenate them. Then we can juts express T as the concatenated list of Tv_1, Tv_2,\dots, Tv_n. We’ll write it like

(Tv_1, Tv_2,\dots, Tv_n)_{\mathcal B\to\mathcal C},

where the subscripts tell us that we’re using the basis \mathcal B for V and \mathcal C for W. If we actually write out the entries Tv_i in their column form, we get something like this:

\left(\begin{array}{ccc}a_{1,1}&\cdots&a_{1,n}\\\vdots&\ddots&\vdots\\a_{m,1}&\cdots&a_{m,n}\end{array}\right)_{\mathcal B\to\mathcal C}.

Look familiar? Matrices are representations of linear transformations in a given basis! It will be important to remember that the bases are largely irrelevant, and that two maps are the same if they do the same thing to the space, even if they look different in different bases. For now, this is a side issue, but know that it will become an important question later on. Tomorrow we will look at lots of examples and how to apply matrices to vectors.

A note on the subscripts:

The notation I am using is 100%  made up by me. Some people have do it like _{\mathcal B}M_{\mathcal C}. I actually like this notation better, but I can’t seem to get the right-side subcripts to work in these posts on a big matrix. Oh well.

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