Matrix Representations

We have seen that linear transformations whose domain and codomain are vector spaces of columns vectors have a close relationship with matrices (, ). In this section, we will extend the relationship between matrices and linear transformations to the setting of linear transformations between abstract vector spaces.

1. fundamental
2. \ltdefn{T}{\complex{2}}{\complex{2}},\quad\lt{T}{\colvector{x_1\\x_2}}=\colvector{2x_1-x_2\\3x_1+2x_2} B&=\set{\colvector{2\\3},\,\colvector{-1\\2}}& C&=\set{\colvector{1\\0},\,\colvector{1\\1}}
3. surprise,

This is a fundamental definition.

We may choose to use whatever terms we want when we make a definition. Some are arbitrary, while others make sense, but only in light of subsequent theorems. Matrix representation is in the latter category. We begin with a linear transformation and produce a matrix. So what? Here is the theorem that justifies the term matrix representation.

This theorem says that we can apply $T$ to $\vect{u}$ and coordinatize the result relative to $C$ in $V$, or we can first coordinatize $\vect{u}$ relative to $B$ in $U$, then multiply by the matrix representation. Either way, the result is the same. So the effect of a linear transformation can always be accomplished by a matrix-vector product (). That is important enough to say again. The effect of a linear transformation is a matrix-vector product. Fundamental Theorem of Matrix Representations \matrix (m) [matrix of math nodes, row sep=5em, column sep=10em, text height=1.5ex, text depth=0.25ex] { \vect{u} & \lt{T}{\vect{u}} \\ \vectrep{B}{\vect{u}} & \matrixrep{T}{B}{C}\,\vectrep{B}{\vect{u}}=\vectrep{C}{\lt{T}{\vect{u}}}\\}; \path[->] (m-1-1) edge[thick] node[auto] {$T$} (m-1-2) (m-1-2) edge[thick] node[auto] {$\vectrepname{C}$} (m-2-2) (m-1-1) edge[thick] node[auto] {$\vectrepname{B}$} (m-2-1) (m-2-1) edge[thick] node[auto] {$\matrixrep{T}{B}{C}$} (m-2-2);

Here is an example to illustrate how the action of a linear transformation can be effected by matrix multiplication.

We will use frequently in the next few sections. A typical application will feel like the linear transformation $T$ commutes with a vector representation, $\vectrepname{C}$, and as it does the transformation morphs into a matrix, $\matrixrep{T}{B}{C}$, while the vector representation changes to a new basis, $\vectrepname{B}$. Or vice-versa.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

In we built new linear transformations from other linear transformations. Sums, scalar multiples and compositions. These new linear transformations will have matrix representations as well. How do the new matrix representations relate to the old matrix representations? Here are the three theorems.

The vector space of all linear transformations from $U$ to $V$ is now isomorphic to the vector space of all $m\times n$ matrices.

This is the second great surprise of introductory linear algebra. Matrices are linear transformations (functions, really), and matrix multiplication is function composition! We can form the composition of two linear transformations, then form the matrix representation of the result. Or we can form the matrix representation of each linear transformation separately, then multiply the two representations together via . In either case, we arrive at the same result.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

It will not be a surprise to discover that the kernel and range of a linear transformation are closely related to the null space and column space of the transformation's matrix representation. Perhaps this idea has been bouncing around in your head already, even before seeing the definition of a matrix representation. However, with a formal definition of a matrix representation (), and a fundamental theorem to go with it () we can be formal about the relationship, using the idea of isomorphic vector spaces (). Here are the twin theorems.

An entirely similar result applies to the range of a linear transformation and the column space of a matrix representation of the linear transformation.

and can be viewed as further formal evidence for the , though they are not direct consequences.

is meant to suggest and , in addition to their proofs (and so carry the same notation as the statements of these two theorems). The dashed lines indicate a subspace relationship, with the smaller vector space lower down in the diagram. The central square is highly reminiscent of . Each of the four vector representations is an isomorphism, so the inverse linear transformation could be depicted with an arrow pointing in the other direction. The four vector spaces across the bottom are familiar from the earliest days of the course, while the four vector spaces across the top are completely abstract. The vector representations that are restrictions (far left and far right) are the functions shown to be invertible representations as the key technique in the proofs of and . So this diagram could be helpful as you study those two proofs. Kernel and Range Isomorphisms \matrix (m) [matrix of math nodes, row sep=2.5em, column sep=5em, text height=1.5ex, text depth=0.25ex] { & U & V & \\ \krn{T} & & & \rng{T} \\ & \complex{n} & \complex{m} & \\ \nsp{\matrixrep{T}{B}{C}} & & & \csp{\matrixrep{T}{B}{C}} \\ }; % linear transformations \path[->] (m-1-2) edge[thick] node[auto] {$T$} (m-1-3) (m-3-2) edge[thick] node[auto] {$\matrixrep{T}{B}{C}$} (m-3-3); % vector representations \path[->] (m-2-1) edge[thick] node[auto] {$\restrict{\vectrepname{B}}{\krn{T}}$} (m-4-1) (m-1-2) edge[thick] node[auto] {$\vectrepname{B}$} (m-3-2) (m-1-3) edge[thick] node[auto] {$\vectrepname{C}$} (m-3-3) (m-2-4) edge[thick] node[auto] {$\restrict{\vectrepname{C}}{\rng{T}}$} (m-4-4); % subsets \path[-] (m-1-2) edge[thick, densely dashed] (m-2-1) (m-1-3) edge[thick, densely dashed] (m-2-4) (m-3-2) edge[thick, densely dashed] (m-4-1) (m-3-3) edge[thick, densely dashed] (m-4-4);

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

We have seen, both in theorems and in examples, that questions about linear transformations are often equivalent to questions about matrices. It is the matrix representation of a linear transformation that makes this idea precise. Here is our final theorem that solidifies this connection.

By now, the connections between matrices and linear transformations should be starting to become more transparent, and you may have already recognized the invertibility of a matrix as being tantamount to the invertibility of the associated matrix representation. The next example shows how to apply this theorem to the problem of actually building a formula for the inverse of an invertible linear transformation.

You might look back at , where we first witnessed the inverse of a linear transformation and recognize that the inverse ($S$) was built from using the method of with a matrix representation of $T$.

This theorem may seem gratuitous. Why state such a special case of ? Because it adds another condition to our NMEx series of theorems, and in some ways it is the most fundamental expression of what it means for a matrix to be nonsingular the associated linear transformation is invertible. This is our final update.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.