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 :
[tex]T(x,y)=(x,y^2).[/tex]
We know that
a transformation T from a vector space U to vector space V is a linear transformation if for [tex]X_1,~X_2[/tex] ∈U and a, b ∈ R
[tex]T(aX_1+bX_2)=aT(X_1)+bT(X_2).[/tex]
So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have
[tex]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)[/tex]
and
[tex]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).[/tex]
Therefore, we get
[tex]T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').[/tex]
Thus, the given transformation T is NOT a linear transformation.
