Linear Algebra

Section II. Homomorphisms 187that for a homomorphism to exist a necessary condition is that the dimension ofthe range must be less than or equal to the dimension of the domain. For instance,there is no homomorphism from R 2 onto R 3 . There are many homomorphismsfrom R 2 into R 3 , but none onto.The range space of a linear map can be of dimension strictly less than thedimension of the domain and so linearly independent sets in the domain maymap to linearly dependent sets in the range. (Example 2.3’s derivative transformationon P 3 has a domain of dimension 4 but a range of dimension 3 and thederivative sends {1, x, x 2 , x 3 } to {0, 1, 2x, 3x 2 }). That is, under a homomorphismindependence may be lost. In contrast, dependence stays.2.19 Lemma Under a linear map, the image of a linearly dependent set is linearlydependent.Proof Suppose that c 1 ⃗v 1 + · · · + c n ⃗v n = ⃗0 V with some c i nonzero. Apply h toboth sides: h(c 1 ⃗v 1 + · · · + c n ⃗v n ) = c 1 h(⃗v 1 ) + · · · + c n h(⃗v n ) and h(⃗0 V ) = ⃗0 W .Thus we have c 1 h(⃗v 1 ) + · · · + c n h(⃗v n ) = ⃗0 W with some c i nonzero. QEDWhen is independence not lost? The obvious sufficient condition is whenthe homomorphism is an isomorphism. This condition is also necessary; seeExercise 35. We will finish this subsection comparing homomorphisms withisomorphisms by observing that a one-to-one homomorphism is an isomorphismfrom its domain onto its range.2.20 Example This one-to-one homomorphism ι: R 2 → R 3⎛ ⎞( ) xx ι ⎜ ⎟↦−→ ⎝y⎠y0gives a correspondence between R 2 and the xy-plane subset of R 3 .2.21 Theorem In an n-dimensional vector space V, these are equivalent statementsabout a linear map h: V → W.(1) h is one-to-one(2) h has an inverse from its range to its domain that is linear(3) N (h) = {⃗0 }, that is, nullity(h) = 0(4) rank(h) = n(5) if 〈⃗β 1 , . . . , ⃗β n 〉 is a basis for V then 〈h(⃗β 1 ), . . . , h(⃗β n )〉 is a basis for R(h)Proof We will first show that (1) ⇐⇒ (2). We will then show that (1) =⇒(3) =⇒ (4) =⇒ (5) =⇒ (2).For (1) =⇒ (2), suppose that the linear map h is one-to-one and so has aninverse h −1 : R(h) → V. The domain of that inverse is the range of h and thusa linear combination of two members of it has the form c 1 h(⃗v 1 ) + c 2 h(⃗v 2 ). On