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)

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)

anonymous · Answer

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.