Question

Let \(W = \left\{ {{{\bf{v}}_1},......,{{\bf{v}}_p}} \right\}\). Show that if \({\bf{x}}\) is orthogonal to each \({{\bf{v}}_j}\), for \(1 \le j \le p\), then \({\bf{x}}\) is orthogonal to every vector in \(W\).

Short answer

It is verified thatif \({\bf{x}}\)is orthogonal to each \({\bf{v}}\) in then \({\bf{x}}\) is orthogonal to each vector in \(W\).

Step-by-step solution

Definition of Orthogonal sets.

The two vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\) are Orthogonal if:

\(\begin{aligned}{l}{\left\| {{\bf{u}} + {\bf{v}}} \right\|^2} &= {\left\| {\bf{u}} \right\|^2} + {\left\| {\bf{v}} \right\|^2}\\{\rm{and}}\\{\bf{u}} \cdot {\bf{v}} &= 0\end{aligned}\)

 Computing the required values.

Since,a vector of \(W\) can be written in the form of linear combination, \({\bf{w}} = {c_1}{{\bf{v}}_1} + ........ + {c_p}{{\bf{v}}_p}\).

As \({\bf{x}}\) orthogonal to vectors \({{\bf{v}}_j}\).

Then, we have:

\(\begin{aligned}{c}{\bf{w}} \cdot {\bf{x}} &= \left( {{c_1}{{\bf{v}}_1} + ........ + {c_p}{{\bf{v}}_p}} \right).{\bf{x}}\\ &= {c_1}\left( {{{\bf{v}}_1} \cdot {\bf{x}}} \right) + ...... + {c_p}\left( {{{\bf{v}}_p} \cdot {\bf{x}}} \right)\\ &= 0 + ....... + 0\\ &= 0\end{aligned}\).

Hence proved,\({\bf{x}}\)is orthogonal to each \({\bf{v}}\) in then \({\bf{x}}\) is orthogonal to each vector in \(W\).