At the end of the course, each student should be able to…

- Module E: Systems of Linear Equations
**E1:**translate back and forth between a system of linear equations and the corresponding augmented matrix.**E2:**put a matrix in reduced row echelon form.**E3:**compute the solution set for a system of linear equations.

- Module V: Vector Spaces
**V1:**explain why a given set with defined addition and scalar multiplication does satisfy a given vector space property, but nonetheless isn’t a vector space.**V2:**determine if a Euclidean vector can be written as a linear combination of a given set of Euclidean vectors.**V3:**determine if a set of Euclidean vectors spans \(\mathbb{R}^n\).**V4:**determine if a subset of \(\mathbb{R}^n\) is a subspace or not.**V5:**determine if a set of Euclidean vectors is linearly dependent or independent.**V6:**determine if a set of Euclidean vectors is a basis of \(\mathbb{R}^n\).**V7:**compute a basis for the subspace spanned by a given set of Euclidean vectors.**V8:**compute the dimension of a subspace of \(\mathbb{R}^n\).**V9:**compute a basis for the subspace spanned by a given set of polynomials or matrices.**V10:**find a basis for the solution set of a homogeneous system of equations.

- Module A: Algebraic Properties of Linear Maps
**A1:**determine if a map between vector spaces of polynomials is linear or not.**A2:**translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.**A3:**compute a basis for the kernel and a basis for the image of a Euclidean linear map.**A4:**determine if a given Euclidean linear map is injective and/or surjective.

- Module M: Matrices
**M1:**multiply matrices.**M2:**describe the row reduction of a matrix as matrix multiplication.**M3:**determine if a square matrix is invertible or not.**M4:**compute the inverse matrix of an invertible matrix.

- Module G: Geometric Properties of Linear Maps
**G1:**describe how a row operation affects the determinant of a matrix.**G2:**compute the determinant of a \(4\times 4\) matrix.**G3:**find the eigenvalues of a \(2\times 2\) matrix.**G4:**find a basis for the eigenspace of a \(4\times 4\) matrix associated with a given eigenvalue.

Additional topics extending the Theory and Applications of linear algebra are covered by the **TA** standard.