Mathematical Immaturity

2.4 Exercises

In each of Exercises 24 through 27, a transformation $T: V \rightarrow V$ is described as indicated. In each case, determine whether $T$ is linear. If $T$ is linear, describe its null space and range, and compute the nullity and rank when they are finite.

25. $\quad$ Let $V$ be the linear space of all real functions differentiable on the open interval $(-1, 1)$. If $f \in V$, $g = T(f)$ means that $g(x) = x f'(x)$ for all $x$ in $(-1, 1)$.

Solution. $\quad$ Recall from Section 2.1 that if $V$ and $W$ are linear spaces, then a function $T: V \to W$ is called a linear transformation from $V$ into $W$ if it has the following two properties:

$\quad$ (a) $\quad T(x + y) = T(x) + T(y) \quad$ for all $x$ and $y$ in $V,$
$\quad$ (b) $\quad T(cx) = cT(x) \quad$ for all $x$ in $V$ and all scalars $c.$

$\quad$ Let $f$ and $g$ be real functions differentiable on the open interval $(-1, 1).$ Then, their sum $f + g$ is also differentiable on the open interval $(-1, 1),$ with $(f + g)' = f' + g',$ which gives us: \begin{align*} T(f + g) &= x(f+g)'(x) \\ &= xf'(x) + xg'(x) \\ &= T(f) + T(g) \end{align*} If $a$ is a scalar, then the function $af$ is also differentiable on the open interval $(-1, 1)$ and we have: \begin{align*} T(af) &= xaf'(x) \\ &= aT(f) \end{align*} As we can see, the transformation $T(f) = xf'(x)$ for all $x$ on the open interval $(-1, 1)$ is linear.

$\quad$ The null space of $T$ is the set of all real functions $f$ differentiable on the open interval $(-1, 1)$ such that $xf'(x) = 0$ for all $x \in (-1, 1).$ In other words, $N(T)$ is the set of all constant $f,$ spanned by $\{1\}.$ The range of $T$ is the set of all real functions $f$ differentiable on the open interval $(-1, 1)$ A subset of which, for example, is the family of functions $e^{nx}$ for $n = 1, 2, \dots.$ As such, $T$ has nullity $1$ and rank $\infty. \quad \blacksquare$