Suppose that T : R3 → R2 is given by:T(a b c) = (a b)a. Prove that T is a linear transformation.b. Find the matrix A such that T(x) = Ax.

Answer:  The required answers are(a) T is proved to be a linear transformation.(b) The matrix A such that T(x) = Ax is [tex]\begin{pmatrix}1 & 0 &0 \\ 0 & 1 &0 \end{pmatrix}[/tex]Step-by-step explanation:  We are given a linear transformation T : R³ → R² defined as follows :[tex]T(a,b,c)=(a,b).[/tex]We are to (a) prove that T is a linear transformationand(b) find a matrix A such that T(x) = Ax.(a) Let s, t are any real numbers and (a, b, c), (a', b', c') ∈ R³.Then, we have[tex]T(s(a,b,c)+t(a',b',c'))\\\\=T(sa+ta',sb+tb',sc+tc')\\\\=(sa+ta',sb+tb')\\\\=(sa,sb)+(ta'+tb')\\\\=s(a,b)+t(a',b')\\\\=sT(a,b,c)+tT(a',b',c').[/tex]So, we get[tex]T(s(a,b,c)+t(a',b',c'))=sT(a,b,c)+tT(a',b',c').[/tex]Therefore, T is a linear transformation.(b) We know that B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} is a standard basis for R³ and B' = {(1, 0), (0, 1)} is a standard basis for R².So, we have[tex]T(1,0,0)=(1,0)=1(1,0)+0(0,1),\\\\T(0,1,0)=(0,1)=0(1,0)+1(0,1),\\\\T(0,0,1)=(0,0)=0(1,0)+0(0,1).[/tex]So, the matrix A such that T(x) = Ax will be given by [tex]\begin{pmatrix}1 & 0 &0 \\ 0 & 1 &0 \end{pmatrix}[/tex]Thus,(a) T is proved to be a linear transformation.(b) The matrix A such that T(x) = Ax is  [tex]\begin{pmatrix}1 & 0 &0 \\ 0 & 1 &0 \end{pmatrix}[/tex]