Numerical Analysis

We will now review some ideas from linear algebra. Proofs of the theorems are either left as exercises or can be found in any standard text on linear algebra. We know how to solve n linear equations in n unknowns. It was assumed that the determinant of the matrix was nonzero and hence that the solution was unique. In the case of a homogeneous system AX = 0 , if , the unique solution is the trivial solution X = 0 . If , there exist nontrivial solutions to AX = 0 . Suppose that , and consider solutions to the homogeneous linear system A homogeneous system of equations always has the trivial solution . Gaussian elimination can be used to obtain the reduced row echelon form which will be used to form a set of relationships between the variables, and a non-trivial solution. Definition (Linearly Independent): The vectors are said to be linearly independent if the equation implies that . If the vectors are not linearly independent they are said to be