Question

Question: 14. Show that if \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is a basis for \({\mathbb{R}^3}\), then aff \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is the plane through \({{\rm{v}}_{\rm{1}}}{\rm{, }}{{\rm{v}}_{\rm{2}}}\) and \({{\rm{v}}_{\rm{3}}}\).

Short answer

It is shown that aff \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is the plane through \({{\rm{v}}_{\rm{1}}}{\rm{, }}{{\rm{v}}_{\rm{2}}}\) and \({{\rm{v}}_{\rm{3}}}\).

Step-by-step solution

Describe the given statement

It is given that \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is in \({\mathbb{R}^3}\) , and it is also known that span of \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) is in the plane \({\mathbb{R}^3}\). Let denote it by \(W\).

 Use linear transformation theory

If \(W = {\rm{span}}\left\{ {{{\bf{v}}_{\rm{2}}} - {{\bf{v}}_{\rm{1}}},{{\bf{v}}_{\rm{3}}} - {{\bf{v}}_{\rm{1}}}} \right\}\), then the plane \(W\) which consists \({{\rm{v}}_{\rm{1}}}\) is parallel to \(W + {{\rm{v}}_{\rm{1}}}\). As \({{\rm{v}}_{\rm{1}}}\), \({{\bf{v}}_{\bf{2}}}\) and \({{\rm{v}}_{\rm{3}}}\) are the linear combination of each other, \(W + {{\bf{v}}_{\bf{1}}}\) must contain \({{\rm{v}}_{\rm{2}}}\), \({{\bf{v}}_{\bf{3}}}\) and so on.

 Use Theorem 1

According to theorem 1, \(y - {{\bf{v}}_{\bf{1}}}\) are the combination of points \({{\bf{v}}_{\bf{2}}} - {{\bf{v}}_{\bf{1}}},...,{{\bf{v}}_{\bf{p}}} - {{\bf{v}}_{\bf{1}}}\), if the point \(y\) is an affine combination of \({{\bf{v}}_{\bf{1}}}{\bf{,}}...,{{\bf{v}}_{\bf{p}}}\) in \({\mathbb{R}^n}\).

So, the plane \(W + {{\bf{v}}_{\bf{1}}}\) is an affine combination of \({{\bf{v}}_{\bf{1}}}{\bf{,}}{\rm{ }}{{\bf{v}}_{\bf{2}}}{\bf{,}}{\rm{ }}{{\bf{v}}_{\bf{3}}}\) as \(W + {{\bf{v}}_{\bf{1}}}\) are combination of points \(\left\{ {{{\rm{v}}_{\rm{2}}} - {{\rm{v}}_{\rm{1}}},{{\rm{v}}_{\rm{3}}} - {{\rm{v}}_{\rm{1}}}} \right\}\) in \({\mathbb{R}^3}\).

Thus, it is shown that aff \(\left\{ {{{\rm{v}}_{\rm{1}}}{\rm{,}}{{\rm{v}}_{\rm{2}}}{\rm{,}}{{\rm{v}}_{\rm{3}}}} \right\}\) is the plane through \({{\rm{v}}_{\rm{1}}}{\rm{, }}{{\rm{v}}_{\rm{2}}}\) and \({{\rm{v}}_{\rm{3}}}\).