The Unapologetic Mathematician

Mathematics for the interested outsider

Matrices I

Yesterday we talked about the high-level views of linear algebra. That is, we’re discussing the category \mathbf{Vec}(\mathbb{F}) of vector spaces over a field \mathbb{F} and \mathbb{F}-linear transformations between them.

More concretely, now: we know that every vector space over \mathbb{F} is free as a module over \mathbb{F}. That is, every vector space has a basis — a set of vectors so that every other vector can be uniquely written as an \mathbb{F}-linear combination of them — though a basis is far from unique. Just how nonunique it is will be one of our subjects going forward.

Now if we’ve got a linear transformation T:V\rightarrow W from one finite-dimensional vector space to another, and if we have a basis \{f_j\}_{j=1}^{\mathrm{dim}(V)} of V and a basis \{g_k\}_{k=1}^{\mathrm{dim}(W)} of W, we can use these to write the transformation T in a particular form: as a matrix. Take the transformation and apply it to each basis element of V to get vectors T(f_j)\in W. These can be written uniquely as linear combinations

\displaystyle T(f_j)=\sum\limits_{k=1}^{\mathrm{dim}(W)}t_j^kg_k

for certain t_j^k\in\mathbb{F}. These coefficients, collected together, we call a matrix. They’re enough to calculate the value of the transformation on any vector v\in V, because we can write

\displaystyle v=\sum\limits_{j=1}^{\mathrm{dim}(V)}v^jf_j

We’re writing the indices of the components as superscripts here, just go with it. Then we can evaluate T(v) using linearity

\displaystyle T(v)=T\left(\sum\limits_{j=1}^{\mathrm{dim}(V)}v^jf_j\right)=\sum\limits_{j=1}^{\mathrm{dim}(V)}v^jT(f_j)=
\displaystyle=\sum\limits_{j=1}^{\mathrm{dim}(V)}v^j\sum\limits_{k=1}^{\mathrm{dim}(W)}t_j^kg_k=\sum\limits_{k=1}^{\mathrm{dim}(W)}\left(\sum\limits_{j=1}^{\mathrm{dim}(V)}t_j^kv^j\right)g_k

So the coefficients v^j defining the vector v\in V and the matrix coefficients t_j^k together give us the coefficients defining the vector T(v)\in W.

If we have another finite-dimensional vector space U with basis \{e_i\}_{i=1}^{\mathrm{dim}(U)} and another transformation S:U\rightarrow V then we have another matrix

\displaystyle S(e_i)=\sum_{j=1}^{\mathrm{dim}(V)}s_i^jf_j

Now we can compose these two transformations and calculate the result on a basis element

\displaystyle \left[T\circ S\right](e_i)=T\left(S(e_i)\right)=T\left(\sum_{j=1}^{\mathrm{dim}(V)}s_i^jf_j\right)=\sum_{j=1}^{\mathrm{dim}(V)}s_i^jT(f_j)=
\displaystyle=\sum_{j=1}^{\mathrm{dim}(V)}s_i^j\sum\limits_{k=1}^{\mathrm{dim}(W)}t_j^kg_k=\sum\limits_{k=1}^{\mathrm{dim}(W)}\left(\sum_{j=1}^{\mathrm{dim}(V)}t_j^ks_i^j\right)g_k

This last quantity in parens is then the matrix of the composite transformation T\circ S. Thus we can represent the operation of composition by this formula for matrix multiplication.

May 20, 2008 - Posted by | Algebra, Linear Algebra

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