4. Linear Systems of Equations

We will consider the solution of (square) systems of linear equations described in the form

\[Ax=b\]

where \(A\) is an \(n\times n\) matrix, and \(x,b\) are \(n\times 1\) vectors. Such a system admits a unique solution if all columns of \(A\) are linearly independent, or in other words, the matrix \(A\) is invertible. We first consider direct methods, where our general strategy will be to transform this system into an equivalent, but easier system (or systems) to solve.