# MA 237 Standards

Linear Algebra - 2018 Spring

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.