MIT OpenCourseWare
43 minutes
Linear Systems of Equations in Linear Algebra
A linear system can be written as $A\mathbf{x} = \mathbf{b}$, where $A$ is an $m \times n$ matrix. The system has a solution exactly when $\mathbf{b}$ is in the column space of $A$. If $\mathbf{b}$ is not in the column space, there is no solution. When a solution exists, it is the only one exactly when the null space of $A$ holds nothing but the zero vector. That happens when the columns of $A$ are linearly independent, so the rank equals $n$. Otherwise there are infinitely many solutions. To work this out you use augmented matrices, row operations, pivot positions, and free variables. For every $m \times n$ matrix, rank plus nullity equals $n$. These ideas are the base for later topics such as eigenvalues.