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Why did the solution to the first question only have one scalar, and the solution to the second had two scalars? Is it up to the solver's preference?

Question 1:

Define $T : \mathbb{P}_3 \to \mathbb{R}^4$ by $$T(p) = \begin{bmatrix} p(-3) \\ p(-1) \\ p(1) \\ p(3) \end{bmatrix}$$ Show that $T$ is a linear transformation.

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Question 2:

Let $T : \mathbb{P}_2$ -> $\mathbb{P}_4$ be the transformation that maps a polynomial $p(t)$ into the polynomial $p(t) + t^2 p(t)$. Show that $T$ is a linear transformation.

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stumped
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1 Answers1

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When seeking to show that $T$ is a linear transformation, the following are equivalent:

  • $T(\alpha x + \beta y) = \alpha T(x) + \beta T(y)$ for any scalars $\alpha, \beta$ and "vectors" $x,y$.
  • $T(x+y) = T(x) + T(y)$ for any $x,y$ and $T(\alpha x) = \alpha T(x)$ for any $x$ and any scalar $\alpha$.

You should be able to prove the equivalence of these statements.

erfink
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  • Would it matter if I started the proof with the right-hand side? As in $\alpha T(x) + \beta T(y) = T(\alpha x + \beta y)$ and $T(x) + T(y)=T(x+y), \alpha T(x) = T(\alpha x)$ – stumped Apr 13 '17 at 18:26