Represent Subspace as Span

Example

Represent subspace $W=\{(x,y,z):x+2y+3z=0\wedge x+y+z=0\}$

The subspace W is the set of solns of:

$$\{\begin{array}{rl}\text{x+2y+3z=0}& \text{(1)}\\ \text{x+y+z=0}& \text{(2)}\end{array}$$

$⟺$ By subtracting (1) from (2)

$$\{\begin{array}{rl}\text{y+2z=0}& \text{(1)}\\ \text{x+y+z=0}& \text{(2)}\end{array}$$

$⟺$

$$\{\begin{array}{rl}\text{y+2z0}& \text{(1)}\\ \text{x-z=0}& \text{(2)}\end{array}$$

$⟺$

$$\{\begin{array}{rl}\text{y=-2z}& \text{(2)}\\ \text{x=z}& \text{(2)}\\ \text{z=z}& \text{(2)}\end{array}$$

This gives $(x,y,z)=(z,-2z,z)$ and so $W=span(1,-2,1)$