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.

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

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