Linear Algebra

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Subject:

Mathematics and Statistics

Institution Name:

The Saylor Foundation

Collection:

Saylor Foundation

Grade Level:

Post-secondary

Abstract:

This course introduces the student to the study of linear algebra. Practically every modern technology relies on linear algebra to simplify the computations required for internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. Upon completion of this course, the student will be able to: Define and identify linear equations; Write a system of equations in matrix-vector form; Explain the geometric interpretation of a system of linear equations; Solve linear equations using a variety of methods; Define general, particular, and homogeneous solutions; Identify how many solutions a linear system has; Correctly manipulate vectors algebraically and perform matrix-vector and matrix-matrix multiplication; Define linear combination and span; Define and distinguish between singular and nonsingular matrices and calculate a matrix inverse; Define and compute LU decompositions; Relate invertibility of matrices to solvability of linear systems; Define and characterize Euclidean space; Define and compute dot and cross-products; Define and identify vector spaces and subspaces; Define spanning set and determine the span of a set of vectors; Define and verify linear independence; Define basis and dimension; Show that a set of vectors is a basis; Define and compute column space, row space, nullspace, and rank; Define and identify isomorphisms and homomorphisms; Use row and column space to solve linear systems; State the rank-nullity theorem; Define inner product, inner product space, and orthogonality; Interpret inner products geometrically; Define determinants using the permutation expansion; State the properties of determinants, such as that the determinant of the product is the product of the determinants; Compute the determinant using cofactor expansions, row reduction, and Cramer's Rule; Define and compute the characteristic polynomial of a matrix; Define and compute eigenvalues and eigenvectors; Explain the geometric significance of eigenvalues and eigenvectors; Define similarity and diagonalizability; Identify similar matrices; Identify some necessary conditions for diagonalizability. (Mathematics 211; See also: Computer Science 105)

Languages:

English

Material Type:

Assessments, Full Course, Homework and Assignments, Readings, Syllabi, Textbooks, Video Lectures

Media Format:

Text/HTML, Downloadable docs, Video

Conditions of Use:

Creative Commons Attribution-Noncommercial 3.0
You are welcome to share, remix, and adapt this course under the terms of the Creative Commons Attribution 3.0 Unported License; however, many linked materials within this course are copyright of their respective authors/owners and may not be openly-licensed. Please respect the copyright and terms of use associated with each resource.

Copyright Holder:

The Saylor Foundation