Question

Suppose \({\bf{y}}\) is orthogonal to \({\bf{u}}\) and \({\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to every \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\). (Hint: An arbitrary \({\bf{w}}\) in Span \(\left\{ {{\bf{u}},\,{\bf{v}}} \right\}\) has the form \({\bf{w}} = {c_1}{\bf{u}} + {c_2}{\bf{v}}\). Show that \({\bf{y}}\) is orthogonal to such a vector \({\bf{w}}\).)

Short answer

It is verified that \({\bf{y}}\)is orthogonal to every \({\bf{w}}\) in span \(\left\{ {{\bf{u}},{\bf{v}}} \right\}\).

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,an arbitrary \({\bf{w}}\)in span \(\left\{ {{\bf{u}},{\bf{v}}} \right\}\) has the form \({\bf{w}} = {c_1}{\bf{u}} + {c_2}{\bf{v}}\).

As \({\bf{y}}\) orthogonal to vectors \({\bf{u}}{\rm{ and }}{\bf{v}}\).

Then, we have:

\(\begin{aligned}{l}{\bf{y}} \cdot {\bf{u}} = 0\\{\bf{y}} \cdot {\bf{v}} = 0\end{aligned}\)

Now, find \({\bf{w}} \cdot {\bf{y}}\).

\(\begin{aligned}{c}{\bf{w}} \cdot {\bf{y}} = \left( {{c_1}{\bf{u}} + {c_2}{\bf{v}}} \right) \cdot {\bf{y}}\\ &= {c_1}\left( {{\bf{u}} \cdot {\bf{y}}} \right) + {c_2}\left( {{\bf{v}} \cdot {\bf{y}}} \right)\\ &= 0 + 0\\ &= 0\end{aligned}\).

Hence proved: \({\bf{y}}\)is orthogonal to every \({\bf{w}}\) in span \(\left\{ {{\bf{u}},{\bf{v}}} \right\}\).