\(\displaystyle{T}{\left({c}_{{1}}{\left({x}_{{1}},{y}_{{1}}\right)}+{c}_{{2}}{\left({x}_{{2}},{y}_{{2}}\right)}\right)}={c}_{{1}}{T}{\left({x}_{{1}},{y}_{{1}}\right)}+{c}_{{2}}{T}{\left({x}_{{2}},{y}_{{2}}\right)}\)

This is done by a direct computation:

\(\displaystyle{T}{\left({c}_{{1}}{\left({x}_{{1}},{y}_{{1}}\right)}+{c}_{{2}}{\left({x}_{{2}},{y}_{{2}}\right)}\right)}\)

\(\displaystyle={T}{\left({c}_{{1}}{x}_{{1}},{c}_{{2}}{x}_{{2}}\right)}+{\left({c}_{{2}}{x}_{{2}},{c}_{{2}}{x}_{{2}}\right)}\)

\(\displaystyle={T}{\left({c}_{{1}}{x}_{{1}}{c}_{{2}}{x}_{{2}},{c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}\right)}\)

\(\displaystyle={\left({3}{\left({c}_{{1}}{x}_{{1}}+{c}_{{2}}{x}_{{2}}\right)}+{\left({c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}\right)},{2}{\left({c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}\right)},{\left({c}_{{1}}{x}_{{1}}+{c}_{{2}}{x}_{{2}}\right)}-{\left({c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}\right)}\right)}\)

\(\displaystyle={\left({3}{c}_{{1}}{x}_{{1}}+{3}{c}_{{2}}{x}_{{2}}+{c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}},{2}{c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}},{c}_{{1}}{x}_{{1}}+{c}_{{2}}{x}_{{2}}-{c}_{{1}}{y}_{{1}}+{c}_{{2}}{y}_{{2}}\right)}\)

\(\displaystyle={\left({3}{c}_{{1}}{x}_{{1}}+{c}_{{1}}{y}_{{1}},{2}{c}_{{1}}{y}_{{1}},{c}_{{1}}{x}_{{1}}-{c}_{{1}}{y}_{{1}}\right)}+{\left({3}{c}_{{2}}{x}_{{2}}+{c}_{{2}}{y}_{{2}},{2}{c}_{{2}}{y}_{{2}},{c}_{{2}}{x}_{{2}}-{c}_{{2}}{y}_{{2}}\right)}\)

\(\displaystyle={c}_{{1}}{\left({3}{x}_{{1}}+{y}_{{1}},{2}{y}_{{1}},{x}_{{1}}-{y}_{{1}}\right)}+{c}_{{2}}{\left({3}{x}_{{2}}+{y}_{{2}},{2}{y}_{{2}},{x}_{{2}}-{y}_{{2}}\right)}\)

\(\displaystyle={c}_{{1}}{T}{\left({x}_{{1}},{y}_{{1}}\right)}+{c}_{{2}}{T}{\left({x}_{{2}},{y}_{{2}}\right)}\)

Therefore, T is linear.