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Unformatted text preview: call # 13420-2 MATH 601, M-W-F 9:30 a.m., SO 0056 A DAY-BY-DAY LIST OF TOPICS Sept. 24: VECTOR SPACES (real & complex), spaces of arrow vectors, the numerical spaces IK n , function spaces, subspaces, finite sets of vectors as opposed to unordered systems and ordered systems of vectors, span of a system, spanning systems Sept. 26: DIMENSION, infinite dimensionality of the space of polynomials, linear (in)dependence, elementary modifications, the replacement theorem Sept. 29: BASIS, the algorithm for selecting a basis from a spanning system, extending a linearly independent system in a finite-dimensional space to a basis, the dimension of a subspace, the case where the space and a subspace are of the same dimension Oct. 1: LINEAR OPERATORS, endomorphisms, scalar operators, matrix operators, the derivative as a linear operator, the operator with prescribed values for a fixed basis, linear functionals (dual vectors), Dirac deltas (evaluation functionals), definite integration as a functional, the spaces L ( V,W ) and End V , the dual space V * , composition Oct. 3: COMPONENTS of a vector relative to a basis, the index notation and summing convention, the dual basis, the equalities L (IK q , IK p ) = IK p q including L (IK q , IK) = IK 1 q , the space of row vectors as the dual of the space IK n of column vectors, the matrix of an operator relative to a pair of bases Oct. 6: IMAGE & KERNEL of a linear operator, rank and nullity, injective & surjective operators, linear equations, homogeneous linear equations, the image of T as the space of those y for which the equation Tx = y is solvable, the kernel of T as the space of solutions to the homogeneous linear equation Tx = 0, surjectivity and injectivity in terms of existence and uniqueness of solutions, injectivity and linear independence, surjectivity and the spanning property, the column space as the image of a matrix operator, linear isomorphisms, isomorphisms between two spaces of the same finite dimension (which always exist, and may be equivalently characterized as being just injective, or just surjective), the Fredholm alternative Oct. 8: TEST 1 Oct. 13: RANK VS. NULLITY (the dimension theorem), cosets of a vector subspace, the set of solutions to a linear equation, solving linear equations by Gaussian elimination Oct. 15: INVERSES of linear isomorphisms, reduced row echelon form, use of Gaus- sian elimination for finding the inverse of an invertible square matrix, differentiation vs. integration, the natural isomorphism : V V ** for a finite-dimensional space V , the B ezout theorem and its main consequence (a nonzero polynomial of degree n cannot have more than n roots), the polynomial interpolation formula, the sum of two subsets in a vector space, the case of two subspaces Oct. 17: QUOTIENT SPACES, addition of cosets, the quotient projection operator P : V V/W , functions up to an additive constant (indefinite integrals), piecewise continuous functions up to equality outside a finite set, the dimension formula dim...
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