Question

T :ℝ2 → ℝ2 as, 𝑇 𝑥, 𝑦 = (1, 𝑦) ;is it a linear transformation?

Answer 1

Here given

T :ℝ2 → ℝ2 , is defined as

T(x,y)=(1,y)

Now,

$for \space T((1,0)+(0,))=T((1,1))=(1,1)$

but $T((1,0))+T((0,1))=(1,0)+(1,1)=(2,1)$

Thus for $(1,0),(0,1)\in\Reals^2,\\T((1,0)+(0,1)\ne T((1,0))+T((0,1))$

and T is not a linear transformation.

By definition, a map, $T:V(F)\to U(F) is linear transformation if T(\alpha u+\beta y)=\alpha T(u)+\beta T(y) \forall\alpha, \beta,\in,F \space and u,y\in V$

or

equivalently

$T(u+y)=T(u)+T(y)\forall u,y\in V\\T(\alpha u)=\alpha T(u)\forall \alpha \in F,u\in V$