Linear Algebra and Its Role in the University Curriculum

Linear Algebra

Linear Algebra is included in the curriculum of almost all university technical majors.

The objective of this publication is to highlight the most common topics that are included in its study at the basic and intermediate level at the University.

It usually begins with a quick review of the Gauss and Gauss-Jordan methods that will be used extensively in calculations as shown in the following illustration.

      x + 2y +z         2x +5y +3z      x +2y +2z = = = 3 7 3

~1 2 1 2 5 3 1 2 2 | | | 3 7 3

This method makes calculations easier by performing row reductions in the matrix instead of manipulating the rows that make up the system of linear equations.

Many courses include a brief review of the determinants, their properties and applications. They include also the study of decompositions such as:

A=LU and A=LDLT for symmetric matrices.

Starting from the solution of homogeneous linear systems, the idea of ​​vector spaces and their relationship with the geometry of Rn are introduced and in some courses affine subspaces are included.

Starting from the definition of ​​ the basis of a subspace orthogonal basis are studied and built using Gram-Schmidt method.

Gram-Schmidt Method

Using this method, orthogonal matrices of great importance in numerical methods can be constructed later.

Equations of lines and planes in R3 and hyperplanes in Rn are usually included.

In the study of finite vector spaces, the relationship between matrices and linear transformations is shown. The kernel and the images of linear transformations and their dimensions are calculated.

Perhaps the most important thing to consider in a basic course is the study of eigenvalues ​​and eigenvectors and its applications including matrix diagonalization.

Some courses include the QR algorithm in order to calculate the real eigenvalues of a matrix.

The SVD decomposition with applications in image processing and other fields is covered in some courses. It uses the eigenvectors of the symmetric matrices and A^T A and the eigenvalues of AA^T or A^T A  (these are the same).

All of these topics are covered in the online support I provide. Find an online alegbra tutor.

Jose A. Barreto. M.A.

The university of Texas at Austin

Jose B.
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