I'm new to linear transformations in linear algebra and I can't quite understand how to find out if a transformation is linear.
Any help would be much appreciated!
a) $T:\mathbb{R}^2 \to\mathbb{R}^4$ defined by $T(x, y) = (xy, 0,x, y)$
b) $T:\mathbb R^3 \to\mathbb R$ defined by $T(x, y, z) = x - 3y +2z$
How do find out if they are a linear transformation?
a) Consider $T(2,2) + T(3,3) = (4,0,2,2) + (9,0,3,3) = (13,0,5,5)$
This is not equal to $T(5,5)$ and thus, T is not a linear transformation
b) Consider $T(ax+by)$ where $a,b,$ are scalars and $x = (x_1,x_2,x_3), y = (y_1,y_2,y_3) $
Then:
$T(ax+by) = ax_1+by_1-3ax_2-3by_2+2ax_3+2ay_3$
$=a(x_1-3x_2+2x_3)+b(y_1-3y_2+2y_3)$
$=aT(x)+bT(y)$
Thus, T is a linear transformation