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Linear dependence and dimension
ndimensional complex Euclidean space
30 other sections not shown
absolute value algebraic arbitrary linear transformation arbitrary vector Ax,x Ax,y bilinear bound called chapter commutative complex numbers conjugate consider coordinate system correspondence cyclic subspaces decomposition definition denote diagonal matrix dimensional direct product direct sum dual space easy to verify eigenvalues equation equivalent Ex,y fact follows formation Hence Hermitian transformations idempotent implies inner product invariant factors inverse irreducible polynomials isomorphism linear combination linear functions linear manifold spanned linear trans linearly independent linearly independent vectors Markoff matrix minimal polynomial moreover necessary and sufficient normal transformation o.n. set obtain orthogonal basis orthonormal set pair of vectors paragraph preceding section proof properties prove rational canonical form real numbers relation representation respectively result scalar multiplication Schwartz inequality spectral form sufficient condition supremum tion transformations of rank uniquely determined unitary space unitary transformation Ux,x vector space whence words write zero