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NotesonLectures9-10 (September30,2005)Let

V

and

W

be

two

F-vector

spaces.

Then

there

is

anatural

structure

ofvectorspaceonV W ={(v, w):v V, w W},asfollows

(v1, w1)+(v2, w2):=(v1+v2, w1+w2)(0.0.1)

k(v1, w1):=(kv1, kw1),foranyv1, v2 V,w1, w2 W,k F. It iseasy tocheck that theaxiomsinthedefinitionofavectorspace(definition1.2.4.) hold.

V canbeviewedasthesubspace{(v, 0):v V} ofV W andW asthesubspace{(0, w):w W}.

Assume V is n-dimensional, with a basis BV = {v1, . . . , vn} and W ism-dimensional,withabasisBW ={w1, . . . , wm}. Thenabasis forV Wis given by BVW = {(vi, 0) : 1 i n} {(0, wj) : 1 j m}, whichimpliesthatthedimensionofV W ism +n. LetuscheckthatBVW isindeedabasis.

linearindependence: Considerthelinearrelationn ai(vi, 0)+mj=1bj(0, wj)=i=1(0, 0). Thisimplies(n aivi,mj=1bjwj)=(0, 0)andthereforein=1aivi =i=10 andm = 0. Since {v1, . . . , vn} inV, respectively {w1, . . . , wm}j=1bjwjinW are linearly independent, it follows that ai = 0, for all 1 i n,respectivelybj =0,forall1 j m. Thisprovesthe linear independence.

spanning set: Let (v, w) be an element inV W. Thenv V can benwrittenas a linear combination v = i=1vi because {v1, . . . , vn} spanV.=mSimilarlywcanbewrittenasw j=1wj. Then(v, w)=n ai(vi, 0)+i=1m bj(0, wj).ThisimpliesthatBVW isaspanningsetforV W.j=1

Alsoremarkthat ifV isasubsetofsomeFk andW isasubspaceofanFp,thenV W isasubspaceofFk+p.

AhomeworkexerciseaskedtoshowthatifV andW aresubspacesofthesamevector spaceU, thenV +W ={v+w :v V, w W}andV Ware also subspaces of the same vector spaceU. Note that V W is nota subspace in general. For example, ifV andW are two lines containingthe origin ofR3 (so they are subspaces), V W is the union of the twolines,so it isnotasubspace(therearevectorsv V andw W suchthatv+w /V W). In this exampleV +W is the plane containing the twolines,andV W isjustapoint,theorigin.

Let us show that if V andW are finite dimensional (of dimensions n,respectively

m)

than

dim(V)+dim(W)=dim(V W)+dim(V +W).

Intheexampleabove,thisis1+1=2+0.SinceV W isasubspaceofV (orW), it isfinitedimensional. Assume

itsdimension isp min{n,m}and let {v1, . . . , vp}bea basis forV W.The set {v1, . . . , vp} is linearly independentas a subset ofV, so it can beextendedtoabasisofV,{v1, . . . , vp, vp+1, . . . vn}. Similarlythere isabasis

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2ofW,{v1, . . . , vp, wp+1, . . . , wm}. Theclaimwouldfollowifweshowedthattheset{v1, . . . , vp, vp+1, . . . , vn, wp+1, . . . , wm}wereabasisforV +W.

linearlyindependent:

consider

the

linear

relation

a1v1 + +apvp +. . . anvn +bp+1wp+1 +. . . bmwm = 0. This implies that a1v1 +. . . anvn =

(bp+1wp+1 +. . . bmwm). Call this element x. The left hand side impliesthat x V and the right hand side that x W. So x V W andthereforexcanbeexpressedasa linearcombinationofthebasisvectorsofVW:x=c1v1+ +cpvp. Thena1v1+ +apvp+ap+1vp+1+. . . anvn =c1v1+ +cpvp,andso(a1c1)v1+ +(apcp)vp+ap+1cp+1+. . . ancn =0.Since{v1, . . . , vn}isabasisforV,andinparticularlinearlyindependent,itfollowsthata1 =c1, . . . , ap =cp andap+1 = =an =0.

Goingbacktotherelationwestartedwithandusingap+1 = =an =0, it implies that a1v1 + +apvp +bp+1wp+1 +. . . bmwm = 0. Since{v1, . . . , vp, wp+1, . . . , wm} isabasisofW, itfollowsthata1 = =ap =0andbp+1 = =bm =0. Inconclusionallscalarsinthelinearrelationmustbezero,andtherefore{v1, . . . , vp, vp+1, . . . , vn, wp+1, . . . , wm}is linearlyindependent.

spanningset: exercise.