Question

Linear Transformation: Let T:R^2 ==> R^3.
We know T(1,1)=(1,2,3) and T(1,2)=(3,4,5).
Use linearity to compute. T(3,3)

Answer 1

So, we know the functions are linear, thus:
\[
z=a_1x+b_1y\\
w=a_2x+b_2y\\
v=a_3x+b_3y
\]Given that:
\[
z=1=a_1(1)+b_1(1)\\
z=3=a_1(1)+b_1(2)\implies\\
b_1=2\\
a_1=-2
\]Et al. Try that.
Unless, by linearity of transforms you mean, where \(a, b\in\mathbb{R}^2\):
\[
T(a)+T(b)=T(a+b)
\]Then we may have multiple possible functions, I think.