Any vector in the form $a \vec u + b \vec v$ is called a linear combination of $\vec u$ and $\vec v$
For a solution $\vec x$ to exist, the vector $\vec b$ must be a linear combo of the columns of A
→ pivot position in each column
If the columns of m x n matrix A span $\mathbb{R}^m$ if the reduced row echelon form of A has a pivot position in every column, then:
$A \vec x = \vec b$ is called homogenous if $\vec b = \vec 0$
→ $A \vec x = 0$ always has the trivial solution $\vec x = \vec 0$
→$A \vec x = \vec 0$ has nontrivial solutions if A does not have pivot columns in every row aka more rows than columns ( yields shifted version of $A \vec x = \vec 0$)