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a-basis abstract system bilinear function block calculations called canonical form chapter characteristic polynomial characteristic values characteristic vectors coefficients column operations combinatorially equivalent commutative complex numbers congruent consider converges coordinate system corresponding defined denote described det(Ai determined diagonal elements diagonal matrix dimension elementary matrices elementary matrix equivalence classes equivalence relation Euclidean example Exercise exists following theorem geometric Hence homomorphism idempotent identity inner product space integers inverse isomorphism linear algebra linear combination linear equations linear transformation linearly independent mapping matric representation method minimal polynomial n-tuples nilpotent non-singular matrix non-zero element normal orthogonal basis notation obtain permutation pivot operations plane positive definite postulates properties Prove Theorem quadratic form rank real numbers real symmetric reduced echelon form represented relative result row equivalent row vector scalar multiple Show similar solution span square matrix subspace superdiagonal symmetric matrix systems of linear T-invariant tion unique vector space Verify