1. Determine the existence and uniqueness of the solution of a linear system, and find all solutions by choosing an effective method such as Gaussian elimination, factorization or diagonalization, etc. 2. Test for linear independence of vectors, orthogonality of vectors and vector spaces. 3. Determine the rank, determinant, inverse, Gram-Schmidt orthogonalization and different factorizations of a matrix. 4. Visualize and compute the four fundamental subspaces of a matrix, and identify their relation to systems of linear equations, and find their dimension and basis. 5. Describe the use of mathematical techniques from linear algebra as applied to computer applications. 6. Compute eigenvalues and eigenvectors of a matrix, use them for diagonalizing, taking its powers, and applying them to solve advanced problems. 7. Identify special properties of a matrix, such as symmetry, positive definiteness, etc., and use this information to facilitate the calculation of matrix characteristics. [Optional Topic] 8. Describe the use of Singular Value Decomposition and Principal Component Analysis in data science algorithms. [Optional Topic]