Linear Algebra

180 Chapter Three. Maps Between Spaces1.35 (a) Where ⃗u,⃗v ∈ R n , by definition the line segment connecting them is the setl = {t · ⃗u + (1 − t) · ⃗v ∣ ∣ t ∈ [0..1]}. Show that the image, under a homomorphismh, of the segment between ⃗u and ⃗v is the segment between h(⃗u) and h(⃗v).(b) A subset of R n is convex if, for any two points in that set, the line segmentjoining them lies entirely in that set. (The inside of a sphere is convex while theskin of a sphere is not.) Prove that linear maps from R n to R m preserve theproperty of set convexity.̌ 1.36 Let h: R n → R m be a homomorphism.(a) Show that the image under h of a line in R n is a (possibly degenerate) line inR m .(b) What happens to a k-dimensional linear surface?1.37 Prove that the restriction of a homomorphism to a subspace of its domain isanother homomorphism.1.38 Assume that h: V → W is linear.(a) Show that the range space of this map {h(⃗v) ∣ ∣ ⃗v ∈ V } is a subspace of thecodomain W.(b) Show that the null space of this map {⃗v ∈ V ∣ ∣ h(⃗v) = ⃗0 W } is a subspace of thedomain V.(c) Show that if U is a subspace of the domain V then its image {h(⃗u) ∣ ∣ ⃗u ∈ U} isa subspace of the codomain W. This generalizes the first item.(d) Generalize the second item.1.39 Consider the set of isomorphisms from a vector space to itself. Is this a subspaceof the space L(V, V) of homomorphisms from the space to itself?1.40 Does Theorem 1.9 need that 〈⃗β 1 , . . . , ⃗β n 〉 is a basis? That is, can we still get awell-defined and unique homomorphism if we drop either the condition that theset of ⃗β’s be linearly independent, or the condition that it span the domain?1.41 Let V be a vector space and assume that the maps f 1 , f 2 : V → R 1 are linear.(a) Define a map F: V → R 2 whose component functions are the given linear ones.( )f1 (⃗v)⃗v ↦→f 2 (⃗v)Show that F is linear.(b) Does the converse hold — is any linear map from V to R 2 made up of twolinear component maps to R 1 ?(c) Generalize.II.2Range space and Null spaceIsomorphisms and homomorphisms both preserve structure. The difference isthat homomorphisms are subject to fewer restrictions because they needn’tbe onto and needn’t be one-to-one. We will examine what can happen withhomomorphisms that cannot happen with isomorphisms.We first consider the effect of not requiring that a homomorphism be ontoits codomain. Of course, each homomorphism is onto some set, namely its range.