- 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
2.8 Exercises
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.
25.$\quad$ 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 $ST \neq TS.$
Solution. $\quad$ Using the definitions of addition, composition (or, multiplication), and integral powers of functions as defined in Section 5, we have: \begin{align*} (S + T)^2 &= (S + T)(S + T) \\ &= S^2 + ST + TS + T^2 \\ \\ (S + T)^3 &= (S + T)(S + T)^2 \\ &= (S + T)(S^2 + ST + TS + T^2) \\ &= (S^3 + S^2T + STS + ST^2) + (TS^2 + TST + T^2S + T^3) \end{align*} If $S$ and $T$ do not commute, the above equations hold. Otherwise, we can use the result of Exercise 22 to give \begin{align*} (S + T)^2 &= S^2 + 2ST + T^2 \\ (S + T)^3 &= S^3 + 3S^2T + 3ST^2 + T^3. \quad \blacksquare \end{align*}