Thursday finish last lecture V - finite dimension vector space / F V' - dual space L (V,F) - set of linear maps; can think of as functions on V - think about pairing - V' x V goes to F - can reverse the roles - given functional phi < > duality between V and V' a new layer is created in linear algebra a canonical isomorphism exists between two things rare highly prized This is one of them Any vector v can be treated as an element of the dual space or Canonical map Theorem: this map D is an isomorphism the dual of the dual Proof: sufficient to show that D is injective, which is equivalent to saying that the null space --- contrast that to Any two vector spaces of dimension n are isomorphic special case in F^n where $\exists$ a canonical basis powerful statement: isomorphism without defining bases all vector spaces come in pairs Dual maps --- numerical representation if choose basis of V dual basis convenient to think of the pairing of the duals as rows on the left, column on right - so the product expresses the relationship as a multiplication - comment on physics notation using b^i to indicate summation over the product: b^i a_i - discussion of duality, reversing the map Use pairing for proof < $\phi$ | T | V > means applying T to V or other way --- Application 36:00 zoo of vector spaces all VS of dimension n have dual - Lagrangian interpolation or polynomial interpolation - use P - dimension n + 1 - define Lagrange polynomials - expression of product $\large\pi_{i\ne j}$ $\frac{t-a_i}{a_a-a_a}$ - check that these polynomials have this property - p sub i of a sub j is the Kroneker delta - Theorem: just knowing linear algebra - notion of dual basis; functionals form basis of V prime - value at point b is always given by combination of values at the points at a sub i with universal coefficients c sub i - for all polynomials - if know polynomial at n+1 points, know its value anywhere - know value of any polynomial anywhere given values at a sub i there is a universal formula cool application One more thing for Dual spaces - linear maps What about subspaces? There must be a subspace in V' subset of all phi in V' such that U ^0 is defined Example V is lambda phi a = valuation at point a. where lambda is in R U zero is all polynomials whose value at a is zero reference to homework problems where had to find values of polynomials not only dimension but define the null space of U dimension is less by 1; term is co-dimension 1 Next result: there is a pattern where dimensions of U and U zero are complementary to each other U zero is subspace, not just subset of V' dimension is = to dimension 3.125 --- 51:14 picture is complete: if T is injective, then null space is zero. annihilator is V' T is injective iff T' is surjective --- Eigenvectors and eigenvalues start now, follow after the exam Preliminaries Short Chapter 4 about polynomials: complex numbers new notation for polynomials Definition: 4.6 4.8 Chapter 4 has proof of theorem that relies on complex analysis not responsible for proof, only result --- eigenvector and eigen value Chapter 5 T is an operator ; still a linear map, but from V to V Example: rotation Suppose you have an operator Two things Subspace U inside V is called invariant under T, or T-invariant if Lemma: suppose you have vi, .... , vk are eigenvectors with eigenvalues that are distinct then this set {v1, ... vk} is linear independent all info on midterm is on bcourses Tuesday review here; exam Thursday