• This web page describes an activity within the Department of Mathematics at Ohio University, but is not an official university web page.
• If you have difficulty accessing these materials due to visual impairment, please email me at mohlenka@ohio.edu; an alternative format may be available.

main calculus sage page

# Linear Algebra

linear algebra quick reference

You can create matrices and scale and add them.

You can multiply matrices.

You can make identity matrices and transpose matrices.

You can compute the norm of a vector

You can add vectors in two dimensions and illustrate the result.

You can add vectors in three dimensions and illustrate the result.

You can solve a system of linear equations by substitution and show the steps.

You can do step-by-step Gaussian elimination.

You can compute the row-reduced form of a matrix. Note that terminology differs between textbooks. If your input has only integers or rational numbers then the result will be expressed in rational numbers; if it includes any decimals (e.g. 1.0) then the result will be expressed in decimals.

You can compute the inverse of a matrix using the row-reduced form or directly.

You can compute pivoted LU decompositions.

You can compute ranks and determinants of matrices.

You can compute the characteristic polynomial, and solve for its zeros.

You can compute the eigenvalues and eigenvectors of a matrix.

You can compute the Jordan canonical form of a matrix, which will be diagonal if the matrix is diagonalizable.

You can compute the exponential of a variable times a matrix. If the eigenvalues are complex, it will look complex.

You can compute the QR decomposition of a matrix.

You can manually compute the QR decomposition of a matrix to illustrate the steps.

You can use the QR algorithm to compute eigenvalues of a matrix. The diagonal should converge to the eigenvalues (under some assumptions).

You can compute the least-squares solution to an overdetermined system.

You can compute various vector and matrix norms.

You can compute the dot product of vectors in any dimension.

You can compute the cross product of vectors in 3 dimensions, and illustrate the result.

Martin J. Mohlenkamp