Determine whether the following are linear transformations from $ℝ^2$ into $ℝ^3$

Question

I don't know how to tackle the next problem, any help is appreciated

Here it is:

Determine whether the following are linear transformations from $ℝ^2$ into $ℝ^3$

a) $L(x)=(x_1,x_2,1)^T$

Well I know I have to check 2 properties,

$L(v_1+v_2)=L(v_1)+L(v_2)$

$L(\alpha v)=\alpha L(v) $ for scalar alpha and vectors in the vector space

my attempt:

$L(\alpha x)=(\alpha x_1,\alpha x_2, \alpha 1)^T=\alpha L(x) $

but the other property I don't know how to do, the 1 isn't like an x (or v)

thanks in advance :)

Answer 1

Suppose $v_1 = (x_1,x_2)^T$ and $v_2 = (y_1,y_2)^T$.

Then $L(v_1 + v_2) = L((x_1,x_2)^T + (y_1,y_2)^T) = L((x_1+y_1,x_2+y_2)^T) = (x_1+y_1,x_2+y_2,1)^T$.

However,

$L(v_1) + L(v_2) = L((x_1,x_2)^T)+L((y_1,y_2)^T) = (x_1,x_2,1)^T + (y_1,y_2,1)^T = (x_1+y_1,x_2+y_2,2).$

Answer 2

We have $$L(v_1) + L(v_2) = (v_{1_1} + v_{2_1},v_{1_2} + v_{2_2},1+1)^T.$$

However $$L(v_1 + v_2) = (v_{1_1} + v_{2_1},v_{1_2} + v_{2_2},1)^T.$$

By the way, I'm not sure you checked the homogeneity condition properly. I think we should get

$$L(\alpha v) = (\alpha x_1, \alpha x_2, 1)^T.$$