Let \(\displaystyle\le{f}{t}{\left\lbrace{v}_{{{1}}},\ {v}_{{{2}}},\dot{{s}},\ {v}_{{{n}}}{r}{i}{g}{h}{t}\right\rbrace}\) be a basis for a vector space V. Prove that if a linear transformation \(\displaystyle{T}\ :\ {V}\rightarrow\ {V}\) satisfies \(\displaystyle{T}{\left({v}_{{{1}}}\right)}={0}\ \text{for}\ {i}={1},\ {2},\dot{{s}},\ {n},\) then T is the zero transformation.

Getting Started: To prove that T is the zero transformation, you need to show that \(\displaystyle{T}{\left({v}\right)}={0}\) for every vector v in V.

(i) Let v be an arbitrary vector in V such that \(\displaystyle{v}={c}_{{{1}}}\ {v}_{{{1}}}\ +\ {c}_{{{2}}}\ {v}_{{{2}}}\ +\ \dot{{s}}\ +\ {c}_{{{n}}}\ {v}_{{{n}}}.\)

(ii) Use the definition and properties of linear transformations to rewrite \(\displaystyle{T}\ {\left({v}\right)}\) as a linear combination of \(\displaystyle{T}\ {\left({v}_{{{1}}}\right)}\).

(iii) Use the fact that \(\displaystyle{T}\ {\left({v}_{{i}}\right)}={0}\) to conclude that \(\displaystyle{T}\ {\left({v}\right)}={0}\), making T the zero tranformation.