Higher Dimension Vector Span (Example)

Example

Question

Compute the linear combinations of the vectors

$$\mathbf{u}=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\text{and }\mathbf{v}=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$$

and show that they span a plane in ${\mathbb{R}}^3$. Find a vector that is not a linear combination of $\mathbf{u}$, $\mathbf{v}$.

Solution

Assume $c_1$ and $c_2$ such that

$$c_1\mathbf{u}+c_2\mathbf{v}⟺\left[\begin{array}{c}{c}_1\\ c_1+c_2\\ c_2\end{array}\right]$$ $$\begin{array}{rl}⟺\text{span}(\mathbf{u},\mathbf{v})& =\left(\left[\begin{array}{c}{c}_1\\ c_1+c_2\\ c_2\end{array}\right]:c_1,c_2\in \mathbb{R}\right)\\ & =\left(\left[\begin{array}{c}x\\ x+z\\ z\end{array}\right]:x,z\in \mathbb{R}\right)\end{array}$$

Hence $y=x+z$, a plane in ${\mathbb{R}}^3$. One vector that contradicts this, is simply any vector that does not lie on this plane: where $y\ne x+z$, e.g.

$$w=\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$$