Span: Examples

Example

Find the span of the vector \(\Vect{u} = (1,1,0)\) in \(\RNrSpc{3}\).

Solution

The span of \(\Vect{u}\) is the collection of all vectors of the form

\(\Vect{x}\)\(=\)\(t\cdot (1,1,0)\)

This is the line in \(\RNrSpc{3}\) passing through the origin and in the direction of \(\Vect{u}\).

Example

Find the span of a single vector \(\Vect{s}\) in \(\RNrSpc{n}\).

Solution

The span of \(\Vect{s}\) is the collection of all vectors of the form

\(\Vect{x}\)\(=\)\(t\cdot \Vect{s}\)

To describe the outcome of this operation we distinguish two cases:

  1. \(\Vect{s} = \Vect{0}\): in this case \(t\cdot \Vect{s} = \Vect{0}\) and, therefore, \(\span(\Vect{s})\) is just the \(\Vect{0}\)-vector.
  2. \(\Vect{s}\neq \Vect{0}\): in this case \(\span(\Vect{s})\) is the line through the origin in the direction of \(\Vect{s}\).

Example

Find the span of the vectors \(\Vect{u}=(2,0)\) and \(\Vect{v}=(0,3)\) in \(\RNrSpc{2}\).

Solution

The span of \(\Vect{u}\) and \(\Vect{v}\) consists of all vectors of the form

\(\Vect{x}\)\(=\)\(s\cdot \Vect{u} + t\cdot \Vect{v}\)
\(\)\(=\)\(s\cdot (2,0)\ +\ t\cdot (0,3)\)
\(\)\(=\)\((2s,3t)\)

We conclude that \(\span(\Vect{u},\Vect{v})\) is all of \(\RNrSpc{2}\) because an arbitrary vector \((a,b)\) in \(\RNrSpc{2}\) is of the form

\((a,b)\)\(=\)\((2\cdot\tfrac{a}{2},3\cdot\tfrac{b}{3})\)

So it is in the span of \(\Vect{u}\) and \(\Vect{v}\).