## Advanced Linear Algebra |

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### Contents

Vector Spaces | 1 |

Linear Transformations | 41 |

Polynomials | 77 |

Copyright | |

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addition and scalar algebra Ann(T arbitrary block diagonal characteristic polynomial cnvn coefficient complex inner product complex numbers compute consequence coordinate vector Corollary defined Definition deg(f(x denote det(A det(B det(T diagonal matrix dimension dimensional inner product dimensional vector space divides f(x dot product eigenvalue eigenvector element Equation Example Exercises exists an orthonormal field F finite dimensional inner finite dimensional vector following hold Gram-Schmidt process implies inductive hypothesis inner product space invertible irreducible polynomial isometry isomorphic Ker(T Lemma Let f(x linear combination linear operator linear transformation linearly independent minimal polynomial monic monic irreducible polynomials Mt(B n x n matrix n—dimensional need to show one-to-one orthogonal set orthonormal basis pi(x polynomial f(x Projw Proof Range(T real inner product real numbers relatively prime respect result scalar multiplication sequence of vectors space with basis Span(S Span{v square root subset Suppose surjective T—invariant subspace T(vn Theorem