Mathematical Immaturity

2.4 Exercises

In each of Exercises 1 through 10, a transformation $T: V_2 \rightarrow V_2$ is defined by the formula given for $T(x, y),$ where $(x, y)$ is an arbitrary point in $V_2$. In each case determine whether $T$ is linear. If $T$ is linear, describe its null space and range, and compute its nullity and rank.

• $T(x, y) = (y, x).$
• $T(x, y) = (x, -y).$
• $T(x, y) = (x, 0).$
• $T(x, y) = (x, x).$
• $T(x, y) = (x^2, y^2).$
• $T(x, y) = (e^x, e^y).$
• $T(x, y) = (x, 1).$
• $T(x, y) = (x + 1, y + 1).$
• $T(x, y) = (x - y, x + y).$
• $T(x, y) = (2x - y, x + y).$

Do the same as above for each of Exercises 11 through 15 if the transformation $T: V_2 \rightarrow V_2$ is described as indicated.

• $T$ rotates every point through the same angle $\phi$ about the origin. That is, $T$ maps a point with polar coordinates $(r, \theta)$ onto the point with polar coordinates $(r, \theta + \phi)$, where $\phi$ is fixed. Also, $T$ maps $0$ onto itself.
• $T$ maps each point onto its reflection with respect to a fixed line through the origin.
• $T$ maps every point onto the point $(1, 1).$
• $T$ maps each point with polar coordinates $(r, \theta)$ onto the point with polar coordinates $(2r, \theta)$. Also, $T$ maps $O$ onto itself.
• $T$ maps each point with polar coordinates $(r, \theta)$ onto the point with polar coordinates $(r, 2\theta)$. Also, $T$ maps $O$ onto itself.

Do the same as above in each of Exercises 16 through 23 if a transformation $T: V_3 \rightarrow V_3$ is defined by the formula given for $T(x, y, z)$, where $(x, y, z)$ is an arbitrary point of $V_3$.

• $T(x, y, z) = (z, y, x).$
• $T(x, y, z) = (x, y, 0).$
• $T(x, y, z) = (x, 2y, 3z).$
• $T(x, y, z) = (x, y, 1).$
• $T(x, y, z) = (x + 1, y + 1, z - 1).$
• $T(x, y, z) = (x + 1, y + 2, z + 3).$
• $T(x, y, z) = (x, y^2, z^3).$
• $T(x, y, z) = (x + z, 0, x + y).$

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.

• Let $V$ be the linear space of all real polynomials $p(x)$ of degree $\leq n$. If $p \in V$, $q = T(p)$ means that $q(x) = p(x + 1)$ for all real $x$.

• 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 $(-1, 1)$.

• 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*}

• Let $V$ be the space of all real functions twice differentiable on an open interval $(a,b).$ If $y\in V,$ define $T(y)=y'' + P\,y' + Q\,y,$ where $P$ and $Q$ are fixed constants.

• Let $V$ be the linear space of all real convergent sequences $\{x_n\}.$ Define a transformation $T:V\to V$ as follows: If $x=\{x_n\}$ is a convergent sequence with limit $a,$ let $T(x)=\{y_n\},$ where $y_n = a - x_n$ for $n\ge1.$ Prove that $T$ is linear and describe the null space and range of $T.$

• Let $V$ denote the linear space of all real functions continuous on $[-\pi,\pi].$ Let $S$ be the subset of $V$ consisting of all $f$ satisfying \begin{align*} \int_{-\pi}^{\pi} f(t)\,dt &= 0,\quad \int_{-\pi}^{\pi} f(t)\cos t\,dt = 0,\quad \int_{-\pi}^{\pi} f(t)\sin t\,dt = 0. \end{align*}
  (a) $\quad$ Prove that $S$ is a subspace of $V.$
  (b) $\quad$ Prove that $S$ contains the functions $f(x)=\cos nx$ and $f(x)=\sin nx$ for each $n=2,3,\dots.$
  (c) $\quad$ Prove that $S$ is infinite-dimensional.
  
  Let $T:V\to V$ be the linear transformation defined as follows: If $f\in V,$ $g=T(f)$ means that \begin{align*} g(x) &= \int_{-\pi}^{\pi}\{1+\cos(x - t)\}f(t)\,dt. \end{align*} (d) $\quad$ Prove that $T(V),$ the range of $T,$ is finite-dimensional and find a basis for $T(V).$
  (e) $\quad$ Determine the null space of $T.$
  (f) $\quad$ Find all real $c\neq0$ and all nonzero $f\in V$ such that $T(f)=c\,f.$ (Note that such an $f$ lies in the range of $T.$)

• Let $T:V\to W.$ be a linear transformation of a linear space $V$ into a linear space $W.$ If $V$ is infinite-dimensional, prove that at least one of $T(V)$ or $N(T)$ is infinite-dimensional.

  [Hint: Assume $\dim N(T)=k,$ $\dim T(V)=r,$ let $e_1,\dots,e_k$ be a basis for $N(T)$ and let $e_{k+1},\dots,e_{k+n}$ be independent elements in $V,$ where $n>r.$ The elements $T(e_{k+1}),\dots,T(e_{k+n})$ are dependent since $n>r.$ Use this fact to obtain a contradiction.]