- 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.
24. $\quad$ 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 $x$.
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 $c.$
$\quad$ Suppose $p$ and $q$ are real polynomials of degree $\leq n.$ Then, the sum $(p + q)$ is also a real polynomial of degree $\leq n,$ the transformation of which for all real $x$ is given by: \begin{align*} T[(p + q)(x)] &= (p + q)(x + 1) \\ &= p(x+1) + q(x+1) \\ &= T[p(x)] + T[q(x)] \end{align*} If $a$ is a scalar, then $ap(x)$ is a polynomial of degree $\leq n$ with transformation $$T[ap(x)] = ap(x+1) = aT[p(x)]$$ for all real $x.$ Thus, we have shown that $T(p) = p(x + 1)$ for all real $x$ is a linear transformation.
$\quad$ The null space of $T$ is the set of all real polynomials of degree $\leq n$ such that $T(p) = O$ for all real $x.$ In other words, $N(T) = \{O\},$ giving $T$ nullity $0.$ The range of $T$ is the set of all real polynomials with degree $\leq n.$ As such, $T(V)$ is spanned by the independent set $\{1, x, \dots, x^n\},$ giving $T$ rank $n + 1. \quad \blacksquare$