Assess students' understanding of the spectral theorem for real symmetric matrices: statement and outline of proof, existence of an orthonormal eigenbasis, diagonalization by orthogonal matrices, and implications for multiplicity and eigenspace structure. Tasks include computing orthonormal eigenvectors and orthogonal diagonalizations, analyzing quadratic forms via principal axes, interpreting geometric consequences (classification of conics/quadric surfaces, invariant subspaces, Rayleigh quotient extremal properties), and explaining connections between algebraic and geometric viewpoints.