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For the following maps, determine whether they are linear transformations or not, and present an appropriate proof.

(a) $T : R^ 3 → R^ 2$ given by $T(x_1, x_2, x_3) = (4x^2 _2 + x^2 _3 , x^2 _1 − x_2x_3).$

(b) $T : P_3 → P_3$ given by $T(p(x)) = p(2) + 3x · p'(x)$, where $p'(x)$ denotes the derivative of the polynomial p(x).

I'm certain that part a is a linear transformation as $T$ doesn't preserve scalar multiplication but I can't seem to figure out part b. Where to start is the issue I'm having. Thanks in advance

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Well you just check the definitions. $T:V\to W$ is a linear transformation if, for all $v,w\in V$ and $a\in\mathbb{R}$ (works for general base fields), we have $$T(av+w)=aT(v)+T(w)$$Let's look at (a): Pick $(0,1,0)$ and $(0,-1,0)$. Then $T(0,0,0)=0$ but $T(0,1,0)+T(0,-1,0)=(4,-1,0)+(4,1,0)=(8,0,0)\not=(0,0,0)=T((0,1,0)+(0,-1,0))$. So this is clearly not linear. The issue is obviously the quadratic terms.

As for $b$, is $P_3$ all degree $\leq 3$ real polynomials or all strictly degree $3$ polynomials (if $ax^3+bx^2+cx+d\in P_3$, do you require $a\not=0$?)

Edit: Actually shouldn't matter: Let $p,q\in P_3$ and $a\in\mathbb{R}$. Then \begin{align*}T(ap+q)&=(ap+q)(2)+3x\cdot (ap+q)'(x)=ap(2)+q(2)+3x\cdot (ap'(x)+q'(x))\\&=ap(2)+q(2)+a3x\cdot p'(x)+3x\cdot q'(x)=a(p(2)+3x\cdot p'(x))+q(2)+3x\cdot q'(x)\\&=aT(p)+T(q)\end{align*}So $T$ is linear.

What's going on is that $(a)$, the processes are all nonlinear: squaring isn't linear because you always get cross terms and you lose signs, which isn't linear. However, in $(b)$ you have evaluation at $2$, addition, multiplication by $3x$, and differentiation. All of these processes are linear, so combining all of them in this manner ought to be linear, and as you can see it follows.

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