# Linear Algebra Toolkit

Author(s):
Przemyslaw Bogacki

Przemyslaw Bogacki
Old Dominion University

Description

The Linear Algebra Toolkit is comprised of modules designed to help a linear algebra student learn and practice a basic linear algebra procedure, such as Gauss-Jordan reduction, calculating the determinant, or checking for linear independence.

Technology

The Linear Algebra Toolkit has been written entirely in PERL. Every effort has been made to make it compatible with a broad range of browsers, however, no guarantee can be made that every browser will properly handle this application.

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List of Modules

Systems of linear equations and matrices

• Row operation calculator. Interactively perform a sequence of elementary row operations on the given m x n matrix A.
• Transforming a matrix to row echelon form. Find a matrix in row echelon form that is row equivalent to the given m x n matrix A.
• Transforming a matrix to reduced row echelon form. Find the matrix in reduced row echelon form that is row equivalent to the given m x n matrix A.
• Solving a linear system of equations. Solve the given linear system of m equations in n unknowns.
• Calculate the inverse using row operations. Find (if possible) the inverse of the given n x n matrix A.

Determinants

• Calculating the determinant using row operations. Calculate the determinant of the given n x n matrix A.

Vector spaces

• Linear independence and dependence. Given the set S = {v1, v2, ... , vn} of vectors in the vector space V, determine whether S is linearly independent or linearly dependent.
• Determining if the set spans the space. Given the set S = {v1, v2, ... , vn} of vectors in the vector space V, determine whether S spans V.
• Finding a basis of the space spanned by the set. Given the set S = {v1, v2, ... , vn} of vectors in the vector space V, find a basis for span S.
• Finding a basis of the null space of a matrix. Find a basis of the null space of the given m x n matrix A. (Also discussed: rank and nullity of A.)

Linear transformations

• Finding the kernel of the linear transformation. Find the kernel of the linear transformation L: V→W. (Also discussed: nullity of L; is L one-to-one?)
• Finding the range of the linear transformation. Find the range of the linear transformation L:  V→W. (Also discussed: rank of L; is L onto W?)

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Przemyslaw Bogacki, "Linear Algebra Toolkit," Convergence (April 2006)