Now that we have seen that a linear transformation $$T:V \rightarrow W$$ can be represented by a matrix when $$V$$ and $$W$$ are finite-dimensional vector spaces, is there a good reason to bother with the definition of a linear transformation when we can just work with matrices?

Recall that in our definition of a linear transformation $$T:V \rightarrow W$$, $$V$$ and $$W$$ are not required to be finite-dimensional. When either $$V$$ or $$W$$ is of infinite dimension, representing $$T$$ by a matrix with finite size is not possible. (Note that there is such a thing as an infinite matrix.) Hence, the question that we might want to address is: Are there linear transformations with domain or codomain being infinite dimensional? The answer is “yes”.

Before we get to the infinite-dimension example, let us looking at a special case for motivation.

Let $$P_k$$ denote the vector space of polynomials in $$x$$ of degree $$k$$ with real coefficients for $$k = 1,2,\ldots$$. Hence, $P_k = \{ a_k x^k + a_{k-1}x^{k-1}+\cdots +a_1 x + a_0 : a_k,a_{k-1},\ldots,a_1\in\mathbb{R}\}.$ Note that $$\dim(P_k) = k+1$$ for all $$k$$.

Let $$D_1:P_2 \rightarrow P_1$$ be given by $D(ax^2 + bx + c) = 2ax + b.$ Then $$D$$ is a linear transformation. Indeed, if $$u = a_1x^2 + b_1 x + c_1$$ and $$v = a_2 x^2 + b_2 x + c_2$$, \begin{eqnarray*} D(u + v) & = & D( (a_1+a_2)x^2 + (b_1+b_2)x + (c_1+c_2) ) \\ & = & 2(a_1+a_2)x + (b_1+b_2) \\ & = & 2(a_1x + b_1) + (2a_2x + b_2) \\ & = & D(u + v). \end{eqnarray*}

Furthermore, if $$u = ax^2 + bx + c$$ and $$\gamma \in \mathbb{R}$$, \begin{eqnarray*} D(\gamma u) & = & D( (\gamma a)x^2 + (\gamma b)x + \gamma ) \\ & = & 2 (\gamma a)x + \gamma b \\ &= & \gamma (2ax + b) = \gamma D(u).\end{eqnarray*}

More generally, for each positive integer $$k$$, $$D_k:P_{k+1} \rightarrow P_{k}$$ given by $D(a_{k+1}x^{k+1} + a_k x^k +\cdots + a_1 x + a_0) =(k+1)a_{k+1}x^k + k a_k x^{k-1} + \cdots + a_1$ is a linear transformation.

If you have taken differential calculus before, you probably have recognized that the output of $$D_k$$ is simply the derivative of the input.

We now remove the degree restriction and consider $$P$$, the vector space of all polynomials in $$x$$ with real coefficients. Recall that $$P$$ is infinite dimensional.

If $$D:P \rightarrow P$$ is given by $D( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0) = na_nx^{n-1} + \cdots + a_1$ for all integers $$n \geq 0$$ and $$a_n,\ldots,a_0 \in \mathbb{R}$$, then one can check that $$D$$ is a linear transformation and it cannot be represented by a matrix having finite size.

In fact, differentiation is a linear transformation over more general vector spaces of functions. For instance, we can replace $$P$$ with the vector space of all differentiable functions. Vector spaces of differentiable functions appear quite often in signal processing and advanced calculus.

## Exercises

1. Let $$\Gamma = ( x^2, x, 1)$$ be an ordered basis for $$P_2$$ and let $$\Omega = ( x , 1)$$ be an ordered basis for $$P_1$$. Find $$[D_1]_{\Gamma}^{\Omega}$$.

2. Let $$k$$ be a positive integer. Let $$\Gamma = ( x^{k+1}, x^k, \ldots, x, 1)$$ be an ordered basis for $$P_{k+1}$$ and let $$\Omega = ( x^k , x^{k-1},\ldots,x, 1)$$ be an ordered basis for $$P_k$$. Find $$[D_k]_{\Gamma}^{\Omega}$$.

3. Let $$F$$ denote the set of functions $$\displaystyle\sum_{k = 0}^\infty (\alpha_k \sin( kx) + \beta_k \cos(kx))$$ such that $$\alpha_k,\beta_k \in \mathbb{R}$$, for all $$k = 0,1,\ldots,$$ and only a finite number of the $$\alpha_k$$'s and $$\beta_k$$'s are nonzero. (In other words, only a finite number of terms in the infinite sum could be nonzero.)

1. Show that $$F$$ is an infinite-dimensional vector space. (Hint: Show that $$\sin(x), \sin(2x), \sin(4x),\ldots, \sin(2^ix),\ldots$$ are linearly independent.)

2. Show that if $$f$$ is in $$F$$, then the derivative of $$f$$ is also in $$F$$.