To solve for \(\displaystyle{T}_{{{2}}}{\left({T}_{{1}}{\left({v}\right)}\right)}\) find \(\displaystyle{T}_{{{1}}}{\left({v}\right)}\) use properties of linear transformation.

\(\displaystyle{T}_{{{1}}}{\left({v}\right)}={T}_{{{1}}}{\left({a}{v}_{{{1}}}+{b}{v}_{{{2}}}\right)}\)

\(\displaystyle{T}_{{{1}}}{\left({v}\right)}={a}{T}_{{{1}}}{\left({v}_{{{1}}}\right)}+{b}{T}_{{{1}}}{\left({v}_{{{2}}}\right)}\)

Substitute \(\displaystyle{3}{v}_{{{1}}}+{v}_{{{2}}}\) for \(\displaystyle{T}_{{{1}}}{\left({v}_{{{1}}}\right)}\) and 0 for \(\displaystyle{T}_{{{1}}}{\left({v}_{{{2}}}\right)}\) in the above expression.

\(\displaystyle{T}_{{{1}}}{\left({v}\right)}={a}{\left({3}{v}_{{{1}}}+{v}_{{{2}}}\right)}+{b}{\left({0}\right)}\)

\(\displaystyle={a}{\left({3}{v}_{{{1}}}+{v}_{{{2}}}\right)}\)

Substitute \(\displaystyle{a}{\left({3}{v}_{{{1}}}+{v}_{{{2}}}\right)}\) for \(\displaystyle{T}_{{{1}}}{\left({v}\right)}\) to find \(\displaystyle{T}_{{{2}}}{\left({T}_{{{1}}}{\left({v}\right)}\right)}\) and it is written as,

\(\displaystyle{T}_{{{2}}}{\left({T}_{{{1}}}{\left({v}\right)}\right)}={T}_{{{2}}}{\left({a}{\left({3}{v}_{{{1}}}+{v}_{{{2}}}\right)}\right)}\)

\(\displaystyle{T}_{{{2}}}{\left({T}_{{{1}}}{\left({v}\right)}\right)}={3}{a}{T}_{{{2}}}{\left({v}_{{{1}}}\right)}+{a}{T}_{{{2}}}{\left({v}_{{{2}}}\right)}\)

Substitute \(\displaystyle-{5}{v}_{{{2}}}\) for \(\displaystyle{T}_{{{2}}}{\left({v}_{{{1}}}\right)}\) and \(\displaystyle-{v}_{{{1}}}+{6}{v}_{{{2}}}\) for \(\displaystyle{T}_{{{2}}}{\left({v}_{{{2}}}\right)}\) in the above expression.

\(\displaystyle{T}_{{{2}}}{\left({T}_{{{1}}}{\left({v}\right)}\right)}={3}{a}{\left(-{5}{v}_{{{2}}}\right)}+{a}{\left(-{v}_{{{1}}}+{6}{v}_{{{2}}}\right)}\)

\(\displaystyle-{15}{a}{v}_{{{2}}}-{a}{v}_{{{1}}}+{6}{a}{v}_{{{2}}}\)

\(\displaystyle=-{a}{v}_{{{1}}}\ -\ {9}{a}{v}_{{{2}}}\)

Therefore, the linear transformation \(\displaystyle{\left({T}_{{{2}}}{T}_{{{1}}}\right)}{\left({v}\right)}\) is \(\displaystyle{\left(-{a}{v}_{{{1}}}-{9}{a}{v}_{{{2}}}\right)}\)

Answer:

Hence, the linear transformation \(\displaystyle{\left({T}_{{{2}}}{T}_{{{1}}}\right)}{\left({v}\right)}\) is \(\displaystyle{\left(-{a}{v}_{{{1}}}-{9}{a}{v}_{{{2}}}\right)}\)