System of linear equations and column space of matrix:

Question

If I have a system of equations:
$$a_1x+b_1y+c_1z = d_1$$
$$a_2x+b_2y+c_2z = d_2$$
$$a_3x+b_3y+c_3z = d_3$$
where the coefficients $a_i, b_i, c_i$ and constants $d_i$ are real, then I know that a zero determinant on the coefficient matrix tells us that we have an inconsistent system, and there are either infinitely many solutions or zero solutions. I read that if the column vector $(d_1,d_2,d_3)^T$ lies outside of the "column space" of our coefficient matrix then we have zero solutions (makes sense as the matrix cannot map to this vector). My question is, e.g., for the matrix
$$\begin{bmatrix}5 & 2&3\\1 & 2&1\\3&2&2\end{bmatrix}$$
how can we check if it maps to e.g. $$\begin{bmatrix}4\\2\\1\end{bmatrix}$$
using this column space property? (i.e. what are the actual steps involved?)
Also, can anyone clarify what exactly is meant by the column space?

Answer 1

You seem to already understand what you are asking.

The column space of a matrix is just the set of all linear combinations of the columns of $A$. It's the subspace spanned by those columns.

So, if $d^T$ is contained in this subspace, then the system is consistent for that specific $d^T$, otherwise it is not.

All of this is just another way of asking if row reducing the augmented matrix $[a^T b^T c^T d^T]$ is consistent. It sounds like maybe you have not got to subspaces yet, but they are not that difficult of a concept if you already understand spans.