- Convert a system of linear equations to matrix form, and vice-versa.
- Convert a matrix to an echelon form (REF), and further to the reduced echelon form (RREF).
- Determine pivot columns, pivot positions, and pivots from an REF.
- Answer the existence and uniqueness quetions.
- Solve an REF system by back substitution.
- Perform row operations.
- Perform vector operations in Rn (vector addition, multiplication by scalar)
- Convert a matrix equation to vector equation, and vice-versa.
- Detemine if a given vector is a linear combination of a given set of vectors.
- Perform matrix operations (addition, multiplication, multiplication by scalar, transpose, power, matrix-vector multiplication).
- Determine if a homogeneous system has a nontrivial solution.
- Determine if a matrix is singular.
- Write a solution set in parametric vector form.
- Determine if a set of vectors is linearly independent.
- Find the matrix of a linear mapping Rn --> Rk.
- Determine if a linear mapping is onto.
- Determine if a linear mapping is one-to-one.

- Determine if A is invertible.
- Compute the inverse of a matrix if it exists.
- Work with elementary matrices and know their correspondence to elementary row operations.
- Put a matrix in LU form if it has an LU form.
- Solve a system using LU factorization.
- Calculate a determinant (using cofactors and row operations).
- Know the effect of row and column operations on the determinant.
- Calculate area or volume of a region using determinant.
- Determine if a subset of a vector space is a subspace (three conditions).
- Know how to use vector space properties. (You do not need to memorise the list of axioms for a vector space.)
- Determine if a given set of vectors is a basis for Rn.
- Work with the vector spaces Rkxn , Pk , C[a,b], Ck[a,b].
- Find a basis for a subspace (Given a set of vectors, find a linearly independent subset).
- Determine the dimension of a subspace.
- Find the row space, column space, and null space of a matrix. Determine the dimensions of these spaces. Find bases for these spaces.
- Calculate the rank of a matrix.
- Calculate the coordinates of a vector in a given basis.
- Perform a change of basis.
- Find the matrices that perform changes of bases (change-of-coordinate matrices, they are the inverses of each other).

- Find the eigenvalues of a matrix.
- Find eigenvectors or the eigenspace corresponding to an eigenvector.
- Determine if a matrix is diagonalizable. If so, find a diagonalization.
- Recognize and use inner product notation.
- Compute inner products. Find the length of a vector. Find the angle between two vectors.
- Find the orthogonal projection of a vector onto another vector.
- Know the complementary properties of Row A and Nul A, and of Col A and Nul AT.
- Find the orthogonal complement of a subspace.
- Determine if a set of vectors is orthogonal (or orthonormal). Determine if a matrix is orthogonal.
- Calculate the coordinates of a vector in a given orthogonal basis.
- Find the projection of a vector b onto a subspace given as the span of orthogonal vectors. (Also, onto a subspace given as the span of orthonormal vectors.)
- Use the matrix method to find the projection of b onto a subspace given as a span of orthogonal vectors.
- Solve least squares problems using the normal equations.
- Solve least squares problems with orthogonal matrix.
- Use the Gram-Schmidt method to find an orthonormal basis of a subspace.
- Find a QR factorization of a matrix.
- Find the projection of a vector $\vc{b}$ onto a subspace given as the span of arbitrary vectors.
- Solve least squares problems using the QR factorization.
- Orthogonally diagonalize a given symmetric matrix.