Study Guide for Math 346 Final Exam
Core Topics
The following is a list of the core topics of Math 346 ordered in decreasing order of importance. You
should ensure that you have mastered the ideas and techniques associated with the following topics:
• Systems of linear equations and their solutions.
• Echelon form and reduced echelon form; rank of a matrix.
• Matrix operations and their properties, including transposes and inverses.
• Subspaces of Rn and subspaces of general vector spaces such as spaces of functions or spaces of
matrices.
• The span of a set of vectors, linear independence, and bases for vectors in Rn and in general
vector spaces.
• Dimension of a subspace and the inferences one can draw from knowing the dimension.
• Subspaces associated with a matrix A viewed as a linear transformation:
Col(A) (also written Im(A)),
Nul(A) (also written Ker(A)),
Row(A).
• The determinant of a matrix: computation using row operations and computation using cofactor
expansions, properties of determinants, and the criterion for invertibility.
• Meaning of Eigenvalues and eigenvectors; diagonalizability of a matrix. Computing eigenvalues
and finding their associated eigenvectors.
• Least Squares Method.
• Determining the matrix of a general linear transformation (from V to W ) with respect to ordered
bases for V and W . This includes familiarity with standard bases of Pn and Rn×m , and finding
a basis for the kernel or image of such a transformation.
• Orthogonality and the Gram–Schmidt algorithm for producing orthonormal bases.
• Orthogonal projections.
• The Singular Value Decomposition (SVD).
• The Real Spectral Theorem and Orthogonal Diagonalization.
• Linear dynamical systems and their solutions.
• Orthogonal subspaces and orthogonal complements.
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Topics Not on the Final Exam for Sections P and S
Students in Sections P and S will not see the following topics on the final exam:
• Quadratic forms.
• QR-factorization.
• Projection matrices.
• Reflection matrices.
• Geometric methods for finding eigenbases for 2 × 2 and 3 × 3 projection or reflection matrices.
• Geometric Interpretation of the Determinant.
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