I want to check to see if I am understanding Linear Transformations. I have the following problem that was given to us in lecture to do for practice.
For a vector space $\mathbb{P}_n(t)$ over $\mathbb{R}$. Define $T$ as follows. For $p(t) = a_nt^n + ... + a_{1}t^{1} + a_{0}t^{0}$, then $(Tp)(t) = a_{n-1}t^{n} + a_{n-2}t^{n-1} + ... + a_{1}t^{2} + a_{0}t^{1}$. Is $T$ a Linear Transformation?
By definition. Let $V$ and $W$ be vector spaces over the field $F$. $T$ is a Linear Transformation from $V$ into $W$ if and only if: \begin{equation} T(c\alpha + \beta) = c(T\alpha) + T\beta \end{equation} $\forall \alpha, \beta \in V$ and all scalars $c\in F$.
Now let $p(t), q(t) \in \mathbb{P}_n(t)$, where $p(t)$ is defined above and $q(t) = b_{n}t^{n} + ... + b_{1}t^{1} + b_{0}t^{0}$ and $c\in\mathbb{R}$
We know in the vector space $\mathbb{P}_n(t)$, addition and scalar multiplication are defined as \begin{eqnarray} (p+q)(t) & = & p(t) + q(t)\\ (cp)(t) & = & cp(t) \end{eqnarray}
We want to show: $T(cp+q)(t) = c(Tp)(t) + (Tq)(t)$. Thus: \begin{eqnarray} T(cp+q)(t) & = & T(c(a_nt^n + ... + a_{1}t^{1} + a_{0}t^{0})+(b_{n}t^{n} + ... + b_{1}t^{1} + b_{0}t^{0}))\\ & = & T((ca_{n}+b_{n})t^{n} + (ca_{n-1}+b_{n-1})t^{n-1}+...+(ca_{1}+b_{1})t^{1} + (ca_{0}+b_{0})t^{0})\\ & = & (ca_{n-1}+b_{n-1})t^{n} + (ca_{n-2}+b_{n-2})t^{n-1}+...+(ca_{0}+b_{0})t^{1}\\ & = & c(a_{n-1}t^n + ... + a_{0}t^{1})+(b_{n-1}t^{n} + ... + b_{0}t^{1})\\ & = & c(Tp)(t) + (Tq)(t) \end{eqnarray}
Thus $T$ is a linear transformation. Am I on the right track?
Sorry for the rather long post. I thank you for taking the time to read this post. I greatly appreciate your feedback, comments, questions, concerns, and criticisms. Take Care and have a wonderful day.