Basis and dimension of the solution space

Homework Statement

Find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of equations.

x - 2y + z = 0
y - z + w = 0
x - y + w = 0

The Attempt at a Solution

(a)
[1 -2 1 0] => [1 0 -1 2]
[0 1 -1 1] => [0 1 -1 1]
[1 -1 0 1] => [0 0 0 0]
basis = {<1,0,-1,2>, <0,1,-1,1>}

(b) for b i make
x = z - 2w
y = z - w

would i set w = s and z = t?
if so.
[x] = [ t - 2s] = [1] + [-2]
[y] = [ t - s] = t[1] + s[-1]
[z] = [ t ] = [1] + [0]
[w] = [ s ] = [0] + [1]

so the dimension would be the number of vectors
so dimension = 2?

im uncertain about that dimension part

LCKurtz
Homework Helper
Gold Member
I didn't check your arithmetic, but assuming it is OK try writing your solution like this:

$$\left ( \begin{array}{c}x\\y\\z\\w\end{array}\right) = t\left ( \begin{array}{c}1\\1\\1\\0\end{array}\right) +s\left ( \begin{array}{c}-2\\-1\\0\\1\end{array}\right)$$
and the answer should become apparent.

vela
Staff Emeritus
Homework Helper
$$\begin{bmatrix}x\\y\\z\\w\end{bmatrix} = t\begin{bmatrix}1\\1\\1\\0\end{bmatrix}+s\begin{bmatrix}-2\\-1\\0\\1\end{bmatrix}$$