Course Learning Outcomes:

1. Solve parametrized and unparametrized systems of linear equations using Gaussian elimination and back substitution by applying elementary row operations on an augmented matrix; for parametrized systems also determine the number of solutions as a function of the parameter.
2. Perform basic matrix algebraic operations (addition, scaling, multiplication), and compute and utilize properties of the determinant of a real n x n matrix (including using it to assess whether a matrix is invertible).
3. Explain the mathematical concept of a real vector space, and determine whether a given subset of a vector space is a vector subspace by working with both the usual Euclidean space Rn and other vector spaces.
4. Demonstrate an understanding of linear combination, linear dependence, linear span, basis and dimension by: i) determining whether a given vector in is the linear span of a family of vectors and whether that family of vectors is linearly independent, ii) computing a basis for a given vector space and its dimension.
5. Define a linear mapping between vector spaces and determine if a given mapping is linear.
6. Define the kernel and image of a linear mapping, compute them for a given real matrix, and explain how they are related to the column vectors of that matrix.
7. Define eigenvalues, eigenspaces and eigenvectors for a given vector space endomorphism and compute them for a real n x n matrix.
8. Prove linear algebraic results for general vector spaces with mathematical reasoning and in precise mathematical language, combining concepts such as vector subspaces, linear span, linear independence, linear mapping, and eigenvalues/eigenspaces/eigenvectors.