Find the augmented matrix of a linear system.
Determine if a matrix is in echelon form or reduced echelon form.
Use row operations on a matrix to obtain its reduced echelon form.
Determine the general solution of a linear system and represent it in vector form.
Compute a linear combination of vectors.
Compute the product of a matrix and a vector.
Write a linear system as a matrix equation and solve it.
Apply to economics and network flows.
Determine if a vector is a linear combination of other vectors.
Determine if a set of vectors is linearly independent.
Determine if a given mapping is a linear transformation.
Find the image of a vector under a linear transformation.
Find a vector whose image under a linear transformation is given.
Find the standard matrix of a linear transformation.
Compute sums and products of matrices.
Compute the powers of a matrix.
Compute the transpose of a matrix.
Apply to linear models in business, science, and engineering.
Find the inverse of a matrix.
Find the inverse of a matrix using row operations.
Solve the equation when the coefficient matrix is invertible.
Find a basis for the null space of a matrix.
Find a basis for the column space of a matrix.
Find the dimensions of the null and column spaces of a matrix.
Find the rank of a matrix.
Compute determinants by cofactor expansions.
Compute determinants by row reduction to echelon form.
Use Cramer's rule to solve linear systems.
Compute the area of the image of a parallelogram under a linear mapping.
Determine if a given set is a subspace of a vector space.
Find the kernel of a linear mapping.
Find a matrix given its column space.
Show that a given set is a subspace of a vector space.
Find bases for the row space, column space, and null space of a matrix.
Find the change-of-coordinates matrix.
Apply to difference equations and Markov chains.
Determine if a given real number is an eigenvalue of a matrix.
Find the eigenspace of a matrix corresponding to an eigenvalue.
Find the characteristic polynomial of a matrix.
Find the eigenvalues of a matrix.
Apply to discrete dynamical systems and differential equations.
Diagonalize a matrix, if possible.
Compute the power of a square matrix.
Compute the inner product of two vectors.
Find a unit vector in the direction of a given vector.
Compute the distance between two vectors.
Determine if two vectors are orthogonal.
Determine if a set of vectors is an orthogonal set.
Compute the orthogonal projection of a vector onto a subspace.
Use the Gram-Schmidt process to produce an orthogonal basis for a subspace.
Find a QR factorization of a matrix.
Find a least-squares solution of a linear systems.
Apply to linear models.