I'm preparing for my exam and I am stuck at these two exercises in which I must prove that the given transformatios are linear. I know that a transformation is linear, if it's closed under adition and scalar multiplication.
1) A vector $\mathbf{a}$ is given in $\mathbb{R}^3$. Let the transformation be defined in $A: \mathbb{R}^3\to\mathbb{R}^3$ as $Ax = \langle x,\mathbf{a}\rangle x$
Is this transformation linear?
2) A transformation is defined in $T: \mathbb{R}_2[x]\to\mathbb{R}_2[x]$ as: $(T(p))(x) = x^2p(1/x)$. Prove, that it's linear and find it's transformation matrix in standard basis.
I know that, to find the transformation matrix, I have to find out where the transformation transforms vectors of the standard basis. I saw a similar example, where the values $\{1,x,x^2\}$ were simply inserted into the equation of the transformation, so this must be somewhat similar.
I thank you in advance for all your help.
Hints:
You must be using the notation $\,(t,r)\,$ to denote some inner product in $\,\Bbb R^3\,$ , most probably the usual euclidean one , so using the bilinearity of the inner product:
$$A(x+y):=(x+y,a)a=\left((x,a)+(y,a)\right)a=(x,a)a+(y,a)a=\ldots\\{}\\\forall\,\,k\in\Bbb R\;\;,\;\;A(kx):=(kx,a)a=k(x,a)a=\ldots$$
For (2):
$$T(p(x)+q(x)):=x^2\left(p\left(\frac{1}{x}\right)+q\left(\frac{1}{x}\right)\right)=\ldots\\\forall\,k\in\Bbb R\;\;,\;\;T(kp(x)):=x^2\left(kp\left(\frac{1}{x}\right)\right)=kx^2p\left(\frac{1}{x}\right)\ldots$$