Linear Algebra: An Introduction to Abstract Mathematics
Springer Science & Business Media, 6 déc. 2012 - 237 pages
Based on lectures given at Claremont McKenna College, this text constitutes a substantial, abstract introduction to linear algebra. The presentation emphasizes the structural elements over the computational - for example by connecting matrices to linear transformations from the outset - and prepares the student for further study of abstract mathematics. Uniquely among algebra texts at this level, it introduces group theory early in the discussion, as an example of the rigorous development of informal axiomatic systems.
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GROUPS AND GROUP HOMOMORPHISMS
VECTOR SPACES AND LINEAR TRANSFORMATIONS
REPRESENTATION OF LINEAR TRANSFORMATIONS
INNER PRODUCT SPACES
additive group augmented matrix bijective calculation called Cayley-Hamilton Theorem chapter characteristic polynomial characteristic subspaces codomain coefficients completes the proof complex inner product complex numbers composition of functions Consider constitute a basis coordinate COROLLARY corresponding cyclic subspaces defined definition det(A determinant diagonal differentiation direct sum eigenvalues eigenvectors elements endomorphism equal equation equivalent example Exercise exists factor field finite finite-dimensional vector space formula given group homomorphism Hence Hermitian Hint Hom(V,V injective inner product space integers inverse image isomorphism jth column kernel Lemma linear combination linear system linear transformation linearly independent mathematics morphism nilpotent nonzero notation Note operation orthogonal projection orthonormal basis permutation previous problem properties PROPOSITION quadratic form Rank-Nullity Theorem real inner product real numbers real vector space result roots scalar multiplication Section Show solution spanning set subgroup subset Suppose surjective Theorem tion transition matrix transpose transpositions trivial unique unitary zero map