This is the first of a series of review videos for first semester linear algebra. You compute a product $Ax$ by combining rows of $A$ with $x$, but you should think of it as a linear combination of the columns of $A$.

Row reduction, also called Gaussian elimination, is the key to handling systems of equations. In this video, we go over the algorithm and how we can make a matrix fairly nice (REF) or very nice (RREF).

The rank of a matrix tells you how many solutions there are to $Ax=0$. The reduced row-echelon form of the matrix tells you what those solutions are.

The column space of a matrix is the span of its columns. The dimension of this space is the rank of the matrix, and a basis consists of the pivot columns.

An $m \times n$ matrix can be viewed as a collection of $n$ vectors in $R^m$. They are linearly independent if the rank of the matrix is $n$, and span if the rank is $m$.