Solve systems of linear equations by a variety of methods which may include any of the following: Gaussian elimination, Gauss-Jordan elimination, using an inverse matrix (if applicable), or Cramer’s rule (if applicable).
Determine if a set of vectors has certain properties such as linearly independence, linearly dependence, is a basis, is orthogonal or orthonormal.
Apply standard matrix operations including computing the inverse of an invertible matrix.
Identify invertible matrices using a variety of equivalent conditions.
Evaluate determinants of square matrices using cofactor expansion or row reduction.
Construct bases for a variety of subspaces, which may include any of the following: row space of a matrix, column space of a matrix, null space of a matrix, eigenspace of a matrix, kernel of a linear transformation, or range of a linear transformation.
Calculate the eigenvalues of a square matrix.
Diagonalize a square matrix or determine that it is not diagonalizable.
Construct an orthonormal basis for an inner product space by using the Gram-Schmidt process.
Compute and analyze the Singular Value Decomposition of a matrix; identify properties of the matrix or its associated linear transformation based on the SVD and vice versa.