Determine the Linear Transformation on the plane?
Solution 1:
We are searching a linear transformation definded as follows: $$\mathcal{L}([1,0,1]^T)=[1,1,1]^T$$ And: $$\mathcal{L}([-1,1,0]^T)=[2,1,3]^T$$ Aa you stated, $\mathcal{L}(u_1)$ and $\mathcal{L}(u_2)$ must be in the plane: $x+z=0$ if that linear transformation exist. Consider the first image, in other words the vector $u_2=[1,1,1]^T$. $u_2$. In order to be on the plane has to have coordinates such that $x+z=0$. But, in this case $x=z=1$, so $1+1\neq 0$. Thus, the lonear transformation doesn' exist.