- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
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.
26. $\quad$ Let $V$ be the linear space of all real functions continuous on $[a, b]$. If $f \in V$, $g = T(f)$ means that \begin{align*} g(x) = \int_a^b f(t) \sin (x - t) \, dt \quad \text{for} \quad a \leq x \leq b. \end{align*}
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 $u$ and $v$ be two elements of $V,$ and let $c$ be a scalar. We then have: \begin{align*} T(u + v) &= \int_a^b(u + v)(t)\sin(x - t)\,dt \\ &= \int_a^b u(t)\sin(x - t)\,dt + \int_a^b v(t)\sin(x - t)\,dt \\ &= T(u) + T(v) \\ \\ T(cu) &= \int_a^b cu(t)\sin(x - t)\,dt \\ &= c\int_a^b u(t)\sin(x - t)\,dt \\ &= cT(u) \end{align*} Thus, $T$ is a linear transformation.
$\quad$ The range of $T$ is the set of real functions $g,$ parameterized by $x,$ given by: \begin{align*} g(x) &= \int_a^b f(t)\sin(x - t)\,dt \end{align*} where $f(t)$ is a real function, continuous on $[a, b].$ Recalling the addition formula for the sine function from Volume 1, Theorem 2.3: \begin{align*} \sin(x+y) &= \sin x \cos y + \cos x \sin y \end{align*} as well as the even and odd properties of the sine and cosine: \begin{align*} \cos(-x) &= \cos x, \quad \sin(-x) = -\sin x \end{align*} the above integral relation can be rewritten as: \begin{align*} g(x) &= \int_a^b f(t)\left[\sin x \cos t - \cos x \sin t\right]\,dt \\ &= \sin x \int_a^b f(t)\cos t\,dt - \cos x \int_a^b f(t)\sin t\,dt \end{align*} But, $\int_a^b f(t)\cos t\,dt$ and $\int_a^b f(t)\sin t\,dt$ are definite integrals, and thus can be treated as scalars. As such, we can see that $T(V)$ is spanned by the independent set $\{\cos x, \sin x\},$ giving $T$ rank $2$
$\quad$ The null space of $T$ is the set of all real functions, continuous on $[a, b],$ such that for all $x \in [a, b]$ \begin{align*} \sin x \int_a^b f(t)\cos t\,dt - \cos x \int_a^b f(t)\sin t\,dt &= 0 \end{align*} From Volume 1, Theorems 1.18 (Invariance Under Translation), and 1.19 (Expansion or Contraction of the Interval of Integration), we can make the substitution $y = 2\pi(t - a)/(b - a)$ to give us \begin{align*} g(x) &= \frac{b - a}{2\pi}\sin x \int_0^{2\pi} f(y)\cos (y)\,dy - \frac{b - a}{2\pi}\cos x \int_0^{2\pi} f(y)\sin (y)\,dy \end{align*} In other words, the null space contains all real functions $f(y)$ such that \begin{align*} \int_0^{2\pi} f(y)\cos (y)\,dy &= 0, \quad \int_0^{2\pi} f(y)\sin (y)\,dy = 0 \end{align*} One such family of functions is given by $f(y) = \sin(ny)\cos(ny)$ for $n = 1, 2, \dots.$ But since there are infinitely many such functions, the nullity of $T$ is infinite. $\quad \blacksquare$