[Math] Dimension of solution space of homogeneous system of linear equations

Question

I have the homogeneous system of linear equations

$$
3x_1 + 3x_2 + 15x_3 + 11x_4 = 0,
$$
$$
x_1 − 3x_2 + x_3 + x_4 = 0,
$$
$$
2x_1 + 3x_2 + 11x_3 + 8x_4 = 0.
$$

I have converted to a augmented matrix and row reduced to
$$\begin{bmatrix}1 & 0 & 4 & -3 & 0\\0 & 1 & 1 & 2/3 & 0\\0 & 0 & 0 & 0 & 0\end{bmatrix}$$

And came up with the general solution:
$$\begin{bmatrix}x_1 \\x_2\\x_3\\x_4\end{bmatrix} = s\begin{bmatrix}4 \\-1\\1\\0\end{bmatrix}+ t\begin{bmatrix}-3\\-2/3\\0\\1\end{bmatrix}$$

I know that the basis is:

$$\left\{\begin{bmatrix}4 \\-1\\1\\0\end{bmatrix},\begin{bmatrix}-3\\-2/3\\0\\1\end{bmatrix}\right\}$$

But how do I determine the dimension of the solution space?

Answer 1

The dimension is equal to the number of basis vectors, by definition. In this case that is 2.