THE GRAM-SCHMIDT PROCESS
The Gram-Schmidt Process produces an orthogonal basis for a nonzero subspace of . It is based on the following theorem. Theorem Let be a basis for a nonzero subspace W of and: . Then is an orthogonal basis forRead More
DIAGONALIZATION OF SYMMETRIC MATRICES
What is so special about symmetric matrices? While not every square matrix is diagonalizable, every symmetric matrix can be diagonalized. Diagonal matrices are easier to work with and have many fascinating properties. In addition, every symmetric matrix can be representedRead More
SIMILAR MATRICES
As discussed in the article Matrices of Linear Transformations, the matrix representing a linear transformation depends of the bases chosen for the vector spaces associated with the transformation. For example, let T be the linear transformation from to . SomeRead More
MATRICES OF LINEAR TRANSFORMATIONS
The central proposition in this article is that for every linear transformation from a vector space to another there exists a matrix that can be viewed as a representation of the transformation and the entries of the matrix depend onRead More
COORDINATES AND CHANGE OF BASIS
One may get confused when asked to specify the coordinates of T as depicted below. Figure 1 However, they may think it easy to determine the coordinates of T if instead of Figure 1, they are given the following figure.Read More
DIAGONALIZATION OF A SQUARE MATRIX
Just as the coordinate matrix of a vector depends on the basis chosen for the associated vector space, the matrix of a linear operator on a vector space depends on the basis for the vector space. It is often favorableRead More
FIRST ENCOUNTER WITH EIGENVALUES AND EIGENVECTORS
Consider the matrix , which can be viewed as a matrix representation of a linear operator on , and the vector . It is easy to check that . In this example, the output of the matrix transformation is aRead More
CRAMER’S RULE
This article elaborates on how to solve systems of linear equations by applying a theorem called the Cramer’s rule. This rule is an alternative to the elimination and substitution techniques. At the first stage of applying the rule, we haveRead More
TECHNIQUES FOR SOLVING SYSTEMS OF LINEAR EQUATIONS IN TWO UNKNOWNS
An arbitrary system of m linear equations in n unknowns has the following general form: where x1, x2, …, xn are the unknowns and the subscripted a’s and b’s denote constants. As with the systems of linear equations inRead More
SQUARE-ROOT MATRICES
Suppose that X and A are matrices and A is positive definite. (A is a positive definite matrix if x’Ax > 0 for every nonzero vector x.) The matrix X is called the square-root of A if X2 = A.Read More