Question

Determine whether T : R^2 -->R^2,T((x.y)) = (x,y^2) is a linear transformation

Answer 1

Answer:  No, the given transformation T is NOT a linear transformation.

Step-by-step explanation:  We are given to determine whether the following transformation T : R² --> R² is a linear transformation or not :

$T(x,y)=(x,y^2).$

We know that

a transformation T from a vector space U to vector space V is a linear transformation if for $X_1,~X_2$ ∈U and a, b ∈ R

$T(aX_1+bX_2)=aT(X_1)+bT(X_2).$

So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have

$T(a(x,y)+b(x',y'))\\\\=T(ax+bx',ay+by')\\\\=(ax+bx',(ay+by')^2)\\\\=(ax+bx',a^2y^2+2abyy'+y'^2)$

and

$aT(x,y)+bT(x',y')\\\\=a(x,y)+b(x', y'^2)\\\\=(ax+bx',ay+by')\\\\\neq (ax+bx',a^2y^2+2abyy'+y'^2).$

Therefore, we get

$T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').$

Thus, the given transformation T is NOT a linear transformation.