Tuesday ## Invertible Linear Maps and Isomorphisms Axler 3D presentation is different; Frenkel will emphasize aspects of maps that belong to set theory, not to linear algebra Hierarchy of formal systems. At the basis, set theorem: sets, maps between them.... Properties of maps between sets Above that: Formal systems of Linear algebra Invertibility comes from set theory, not linear algebra. Lives at basic level. Plain sets, without any of the properties of vector spaces. Maps of sets: - With two sets: $S_1$ and $S_2$ : map: a rule that assigns to each element of S1 an element of S2 - Injective: send different elements of S1 to different elements of S2 - Surjective: for every element in S2, there is an element in S1 that maps to it - Bijective: Injective and Surjective - 1-1 correspondence - for every s in one, there is one in the other - equivalent property: - Definition: Invertible - identical to Bijective - if a map g in the opposite direction - composition of maps: - sets S1 to S2 to S3 : map from each next - S1 to S2 is f; S2 to S3 is g - second map now goes from S2 back to S\ - the composition goes from S1 to itself - Identity Map: special map from set to itself - element x in S1 is mapped to x in S1; call this the identity map I - now, reverse the roles of S1 and S2. - Lemma: map from S1 to S2 is invertible iff it is bijective - leave proof as exercise - if 1-1 one way, it's 1-1 - how does invertibility imply bijective - Statement of uniqueness: - have map f that is invertible, if there is g. - could be g and g' - seems g is unique - if g and g' both satisfy the condition of definition of invertible maps they are identical - Argument: of negative or additive inverse; or zero - g. precomposition, - Proof substitutes conditions: sequence of equalities gives equivalence between g and g' - if f is an invertible map, it's inverse is unique - now, special case: both sets are Vector spaces, with operations - T: not random map, but compatible; over same field - now, Linear Map;injective, surjective, bijective; now, use Lemma 1 to say that T is bijective iff T is invertible - since g is unique, now denoted as f inverse; for additive inverse, used -; here, use -1 - T has T^-1 - now, establish that T^-1 is a linear map: lemma 3 - Isomorphism - Special case: V, W finite dimensional - dimension is a criterion: isomorphic between spaces if have same dimension - 60min; quantum theory, Hilbert space, infinite dimensional Hilbert space; vectors - two applications of linear algebra in the last 100 years; quantum theory and machine learning - discussion of choice of basis as determining the collapse of the wave function - 1: 15 ; second application: tokens in natural language as vectors; how is the choice of a basis - Lemma 4; let V, W be finite dimensional vector fields over F - map T is isomorphism iff it is injective - and iff it is surjective - similar: check if set of vectors is basis, check it is LI - spanning falls out - example; not a bijection - linear map from infinite dimensional vector space - infinite collections of reals; infinite sequences of real numbers - addition, scalar multiplication defined component wise - vector space; not spanned by finitely many; cannot get to infinite - map; shift by one - map : shift in opposite direction