Question

Suppose vectors \({{\bf{v}}_{\bf{1}}},...,{{\bf{v}}_{\bf{p}}}\) span\({\mathbb{R}^{\bf{n}}}\), and let \({\bf{T}}:{\mathbb{R}^{\bf{n}}} \to {\mathbb{R}^{\bf{n}}}\) be a linear transformation. Suppose \({\bf{T}}\left( {{{\bf{v}}_{\bf{i}}}} \right) = {\bf{0}}\) for \({\bf{i}} = {\bf{1}},...,{\bf{p}}\). Show that \({\bf{T}}\)is the zero transformation. That is, show that if \({\bf{x}}\) is any vector in \({\mathbb{R}^{\bf{n}}}\), then\({\bf{T}}\left( {\bf{x}} \right) = {\bf{0}}\).

Short answer

It has been shown that \(T\) is the zero transformation.

Step-by-step solution

Use the definition of span

Given, the vectors \({v_1},...,{v_p}\) spans \({\mathbb{R}^n}\).

So., \({\mathbb{R}^n} = \left\{ {{c_1}{v_1} + ... + {c_p}{v_p}:{c_1},...,{c_p} \in \mathbb{R}} \right\}\).

Let \(x \in {\mathbb{R}^n}\).

This implies \(x = {c_1}{v_1} + ... + {c_p}{v_p}\) for some \({c_1},...,{c_p} \in \mathbb{R}\).

Use the properties of a linear transformation

\(\begin{aligned} T\left( x \right) &= T\left( {{c_1}{v_1} + ... + {c_p}{v_p}} \right)\\ &= T\left( {{c_1}{v_1}} \right) + ... + T\left( {{c_p}{v_p}} \right)\\ &= {c_1}T\left( {{v_1}} \right) + ... + {c_p}T\left( {{v_p}} \right)\\T\left( x \right) &= 0\end{aligned}\)

Since \(T\left( {{v_i}} \right) = 0\forall i = 1,2,...,p\).

Conclusion

Therefore, \(T\left( x \right) = 0\) for all \(x \in {\mathbb{R}^n}\). Hence, it can be concluded that \(T\) is a zero transformation.