I am completely lost on these two problems and I would like a step by step explanation. Can you please help me out?
1.) Let T be a transformation defined by $$T(x_{1}, x_{2}) = (4x_{1} - 2x_{2}, 3|x_{2}|)$$ Determine whether T is linear. Justify your answer.
2.) Let T: $$ \mathbb{R}^2 \rightarrow \mathbb{R}^4 $$ be a linear transformation such that
$$ T(x_{1},x_{2}) = (2x_{2} - 3x_{1}, x_{1} - 4x_{2}, 0, x_{2}) $$ (a) Find the standard matrix for the transformation T.
(b) Is T a one-to-one linear transformation? Justify your answer.
(c) Does T map $\mathbb{R}^2$ onto $ \mathbb{R}^4$. Justify your answer.
Any help would be appreciated. Thank you!
I'll get you going with $(1)$. In order for $T$ to be linear, it must satisfy $T(av+bu)=aT(v)+bT(u)$ for all scalars $a,b$ and vectors $v,u$. In this case, for $T$ to be linear, we must have $$T(x_1+y_1,x_2+y_2)=T(x_1,x_2)+T(y_1,y_2)$$ Does this hold true in general? In particular, is $|x_2+y_2|=|x_2|+|y_2|$ true in general?