Mathematical Immaturity

2.8 Exercises

• Let $V = \{0, 1\}.$ Describe all functions $T: V \rightarrow V.$ There are four altogether. Label them as $T_1, T_2, T_3, T_4$ and make a multiplication table showing the composition of each pair. Indicate which functions are one-to-one on $V$ and give their inverses.
• Let $V = \{0, 1, 2\}.$ Describe all functions $T: V \rightarrow V$ for which $T(V) = V.$ There are six altogether. Label them as $T_1, \ldots, T_6$ and make a multiplication table showing the composition of each pair. Indicate which functions are one-to-one on $V$ and give their inverses.

In each of Exercises 3 through 12, a function $T: V_2 \rightarrow V_2$ 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 one-to-one on $V_2.$ If it is, describe its range $T(V_2);$ for each point $(u, v)$ in $T(V_2),$ let $(x, y) = T^{-1}(u, v)$ and give formulas for determining $x$ and $y$ in terms of $u$ and $v.$

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

In each of Exercises 13 through 20, a function $T: V_3 \rightarrow V_3$ is defined by the formula given for $T(x, y, z),$ where $(x, y, z)$ is an arbitrary point in $V_3.$ In each case, determine whether $T$ is one-to-one on $V_3.$ If it is, describe its range $T(V_3);$ for each point $(u, v, w)$ in $T(V_3),$ let $(x, y, z) = T^{-1}(u, v, w)$ and give formulas for determining $x,$ $y,$ and $z$ in terms of $u,$ $v,$ and $w.$

• $T(x, y, z) = (z, y, x).$
• $T(x, y, z) = (x, y, 0).$
• $T(x, y, z) = (x, 2y, 3z).$
• $T(x, y, z) = (x, y, x + y + z).$
• $T(x, y, z) = (x + 1, y + 1, z - 1).$
• $T(x, y, z) = (x + 1, y + 2, z + 3).$
• $T(x, y, z) = (x, x + y, x + y + z).$
• $T(x, y, z) = (x + y, y + z, x + z).$
• Let $T: V \rightarrow V$ be a function which maps $V$ into itself. Powers are defined inductively by the formulas $T^0 = I,$ $T^n = TT^{n-1}$ for $n \geq 1.$ Prove that the associative law for composition implies the law of exponents: $T^m T^n = T^{m+n}.$ If $T$ is invertible, prove that $T^n$ is also invertible and that $(T^n)^{-1} = (T^{-1})^n.$

In Exercises 22 through 25, $S$ and $T$ denote functions with domain $V$ and values in $V.$ In general, $ST \neq TS.$ If $ST = TS,$ we say that $S$ and $T$ commute.

• If $S$ and $T$ commute, prove that $(ST)^n = S^n T^n$ for all integers $n \geq 0.$
• If $S$ and $T$ are invertible, prove that $ST$ is also invertible and that $(ST)^{-1} = T^{-1} S^{-1}.$ In other words, the inverse of $ST$ is the composition of inverses, taken in reverse order.
• If $S$ and $T$ are invertible and commute, prove that their inverses also commute.
• Let $V$ be a linear space. If $S$ and $T$ commute, prove that \begin{align*} (S + T)^2 = S^2 + 2ST + T^2 \quad \text{and} \quad (S + T)^3 = S^3 + 3S^2T + 3ST^2 + T^3. \end{align*} Indicate how these formulas must be altered if $ST \neq TS.$
• Let $S$ and $T$ be the linear transformations of $V_3$ into $V_3$ defined by the formulas $S(x, y, z) = (z, y, x)$ and $T(x, y, z) = (x, x + y, x + y + z),$ where $(x, y, z)$ is an arbitrary point of $V_3.$
  (a) $\quad$ Determine the image of $(x, y, z)$ under each of the following transformations: $ST,$ $TS,$ $ST - TS,$ $S^2,$ $T^2,$ $(ST)^2,$ $(TS)^2.$
  (b) $\quad$ Prove that $S$ and $T$ are one-to-one on $V_3$ and find the image of $(u, v, w)$ under each of the following transformations: $S^{-1},$ $T^{-1},$ $(ST)^{-1},$ $(TS)^{-1}.$
  (c) $\quad$ Find the image of $(x, y, z)$ under $(T - I)^n$ for each $n \geq 1.$
• Let $V$ be the linear space of all real polynomials $p(x).$ Let $D$ denote the differentiation operator and let $T$ denote the integration operator which maps each polynomial $p$ onto the polynomial $q$ given by $q(x) = \int_0^x p(t) \, dt.$ Prove that $DT = I_V$ but that $TD \neq I_V.$ Describe the null space and range of $TD.$
• Let $V$ be the linear space of all real polynomials $p(x).$ Let $D$ denote the differentiation operator and let $T$ be the linear transformation that maps $p(x)$ onto $x p'(x).$
  (a) $\quad$ Let $p(x) = 2 + 3x - x^2 + 4x^3$ and determine the image of $p$ under each of the following transformations: $D,$ $T,$ $DT,$ $TD,$ $DT - TD,$ $T^2 D^2 - D^2 T^2.$
  (b) $\quad$ Determine those $p$ in $V$ for which $T(p) = p.$
  (c) $\quad$ Determine those $p$ in $V$ for which $(DT - 2D)(p) = 0.$
  (d) $\quad$ Determine those $p$ in $V$ for which $(DT - TD)^n(p) = D^n(p).$
• Let $V$ and $D$ be as in Exercise 28 but let $T$ be the linear transformation that maps $p(x)$ onto $x^2 p'(x).$ Prove that $DT - TD = Z$ and that $DT^n - TD^n = n T^{n-1}$ for $n \geq 2.$
• Let $S$ and $T$ be in $\mathscr{L}(V, V)$ and assume that $ST - TS = I.$ Prove that $ST^n - T^n S = n T^{n-1}$ for all $n \geq 1.$
• Let $V$ be the linear space of all real polynomials $p(x).$ Let $R,$ $S,$ $T$ be the functions which map an arbitrary polynomial $p(x) = c_0 + c_1 x + \cdots + c_n x^n$ in $V$ onto the polynomials $r(x),$ $s(x),$ and $t(x),$ respectively, where \begin{align*} r(x) = p(0), \quad s(x) = \sum_{k=1}^n c_k x^{k-1}, \quad t(x) = \sum_{k=0}^n c_k x^{k+1}. \end{align*} (a) $\quad$ Let $p(x) = 2 + 3x - x^2 + x^3$ and determine the image of $p$ under each of the following transformations: $R,$ $S,$ $T,$ $ST,$ $TS,$ $(TS)^2,$ $T^2 S^2,$ $S^2 T^2,$ $TRS,$ $RST.$
  (b) $\quad$ Prove that $R,$ $S,$ and $T$ are linear and determine the null space and range of each.
  (c) $\quad$ Prove that $S$ is one-to-one and determine its inverse.
  (d) $\quad$ If $n \geq 1,$ express $(TS)^n$ and $S^n T^n$ in terms of $I$ and $R.$
• Refer to Exercise 28 of Section 2.4. Determine whether $T$ is one-to-one on $V.$ If it is, describe its inverse.