Linear Algebra

Section I. Isomorphisms 159vector space correspond under an isomorphism. Since we studied vector spacesto study linear combinations, “of interest” means “pertaining to linear combinations”.Not of interest is the way that the vectors are presented typographically(or their color!).As an example, although the definition of isomorphism doesn’t explicitly saythat the zero vectors must correspond, it is a consequence of that definition.1.8 Lemma An isomorphism maps a zero vector to a zero vector.Proof. Where f : V → W is an isomorphism, fix any ⃗v ∈ V . Then f(⃗0 V ) =f(0 · ⃗v) = 0 · f(⃗v) = ⃗0 W . QEDThe definition of isomorphism requires that sums of two vectors correspondand that so do scalar multiples. We can extend that to say that all linearcombinations correspond.1.9 Lemma For any map f : V → W between vector spaces these statementsare equivalent.(1) f preserves structuref(⃗v 1 + ⃗v 2 ) = f(⃗v 1 ) + f(⃗v 2 ) and f(c⃗v) = c f(⃗v)(2) f preserves linear combinations of two vectorsf(c 1 ⃗v 1 + c 2 ⃗v 2 ) = c 1 f(⃗v 1 ) + c 2 f(⃗v 2 )(3) f preserves linear combinations of any finite number of vectorsf(c 1 ⃗v 1 + · · · + c n ⃗v n ) = c 1 f(⃗v 1 ) + · · · + c n f(⃗v n )Proof. Since the implications (3) =⇒ (2) and (2) =⇒ (1) are clear, we needonly show that (1) =⇒ (3). Assume statement (1). We will prove statement (3)by induction on the number of summands n.The one-summand base case, that f(c⃗v 1 ) = c f(⃗v 1 ), is covered by the assumptionof statement (1).For the inductive step assume that statement (3) holds whenever there are kor fewer summands, that is, whenever n = 1, or n = 2, . . . , or n = k. Considerthe k + 1-summand case. The first half of (1) givesf(c 1 ⃗v 1 + · · · + c k ⃗v k + c k+1 ⃗v k+1 ) = f(c 1 ⃗v 1 + · · · + c k ⃗v k ) + f(c k+1 ⃗v k+1 )by breaking the sum along the final ‘+’. Then the inductive hypothesis lets usbreak up the k-term sum.= f(c 1 ⃗v 1 ) + · · · + f(c k ⃗v k ) + f(c k+1 ⃗v k+1 )Finally, the second half of statement (1) gives= c 1 f(⃗v 1 ) + · · · + c k f(⃗v k ) + c k+1 f(⃗v k+1 )when applied k + 1 times.QED