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Jf3 THE nDIMENSIONAL VECTOR SPACE V
_9 HOMOMORPHISMS OF VECTOR SPACES
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1-dimensional subspaces affine plane affine space antisymmetric arbitrary called chapter circle classification cobasis coefficient collinear columns complex numbers conjugate consider coordinate system cross-ratio defined definition denoted determined diagonal dimension dimensional distance dual eigenvalues eigenvector elements ellipse endomorphism equal equation equivalence classes euclidean coordinates euclidean space euclidean vector space exists geometry given Hence hermitian functions homogeneous coordinates homogeneous linear function homomorphism hyperbolic hyperplane identical independent inner product intersection invariant kernel level sets linear functions linear variety linearly dependent main theorem matrix morphism motion multiplication normal form obtained one-to-one mapping orthogonal pair parallel parallelogram permutation perpendicular point set points at infinity polar polynomial problem projective plane projective space Proof properties prove quadratic function quadratic variety quadric rank ratio real affine real numbers real vector space represented respect scalars topological triangle unitary w-dimensional w-space zero