Math 2331 { Linear Algebra

1 Vector Spaces & SubspacesMath 2331 Linear Vector Spaces & SubspacesJiwen HeDepartment of Mathematics, University of jiwenhe/math2331 Jiwen He, University of HoustonMath 2331, Linear Algebra1 / Vector Spaces & SubspacesVector Spaces subspaces Determining Vector Spaces & SubspacesVector Spaces: DefinitionVector Spaces: Examples2 2 matricesPolynomialsSubspaces: DefinitionSubspaces: ExamplesDetermining SubspacesJiwen He, University of HoustonMath 2331, Linear Algebra2 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesVector SpacesMany concepts concerning vectors inRncan be extended to othermathematical can think of avector spacein general, as a collection ofobjects that behave as vectors do inRn. The objects of such a setare SpaceAvector spaceis a nonempty setVof objects, calledvectors, onwhich are defined two operations, calledadditionandmultiplication by scalars(real numbers), subject to the ten axiomsbelow.

2 The axioms must hold for allu,vandwinVand for +vis +v=v+ He, University of HoustonMath 2331, Linear Algebra3 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesVector Spaces (cont.)Vector Space (cont.)3. (u+v) +w=u+ (v+w)4. There is a vector (called the zero vector)0inVsuch thatu+0= For eachuinV, there is vector uinVsatisfyingu+ ( u) = (u+v) =cu+ (c+d)u=cu+ (cd)u=c(du).10. 1u= He, University of HoustonMath 2331, Linear Algebra4 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesVector Spaces: ExamplesExampleLetM2 2={[abcd]:a,b,c,dare real}In this context, note that the0vector is[].Jiwen He, University of HoustonMath 2331, Linear Algebra5 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesVector Spaces: PolynomialsExampleLetn 0 be an integer and letPn= the set of all polynomials of degree at mostn ofPnhave the formp(t) =a0+a1t+a2t2+ +antnwherea0,a1.

3 ,anare real numbers andtis a real variable. ThesetPnis a vector will just verify 3 out of the 10 axioms (t) =a0+a1t+ +antnandq(t) =b0+b1t+ + a He, University of HoustonMath 2331, Linear Algebra6 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesVector Spaces: Polynomials (cont.)Axiom 1:The polynomialp+qis defined as follows:(p+q) (t) =p(t)+q(t). Therefore,(p+q) (t) =p(t)+q(t)= () + ()t+ + ()tnwhich is also aof degree at most. Sop+qis He, University of HoustonMath 2331, Linear Algebra7 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesVector Spaces: Polynomials (cont.)Axiom 4:0=0 + 0t+ + 0tn(zero vector inPn)(p+0) (t)=p(t)+0= (a0+ 0) + (a1+ 0)t+ + (an+ 0)tn=a0+a1t+ +antn=p(t)and sop+0=pJiwen He, University of HoustonMath 2331, Linear Algebra8 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesVector Spaces: Polynomials (cont.)

4 Axiom 6:(cp) (t) =cp(t) = () + ()t+ + ()tnwhich is other 7 axioms also hold, soPnis a vector He, University of HoustonMath 2331, Linear Algebra9 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesSubspacesVector spaces may be formed from subsets of other vectors are a vector spaceVis a subsetHofVthat has threeproperties:a. The zero vector ofVis For eachuandvare inH,u+vis inH.(In this case wesayHis closed under vector addition.)c. For eachuinHand each scalarc,cuis inH.(In thiscase we sayHis closed under scalar multiplication.)If the subsetHsatisfies these three properties, thenHitself is avector He, University of HoustonMath 2331, Linear Algebra10 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesSubspaces: ExampleExampleLetH= a0b :aandbare real.

5 Show thatHis asubspace :Verify properties a, b and c of the definition of The zero vector ofR3is inH(leta=andb=).b. Adding two vectors inHalways produces another vector whosesecond entry isand therefore the sum of two vectors inHisalso inH.(His closed under addition)c. Multiplying a vector inHby a scalar produces another vector inH(His closed under scalar multiplication).Since properties a, b, and c hold,Vis a subspace He, University of HoustonMath 2331, Linear Algebra11 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesSubspaces: Example (cont.)NoteVectors (a,0,b) inHlook and act like the points (a,b) Depiction of HJiwen He, University of HoustonMath 2331, Linear Algebra12 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesSubspaces: ExampleExampleIsH={[xx+ 1]:xis real}a subspace of?

6 , doesHsatisfy properties a, b and c?Solution:ForHto be a subspace ofR2,all three propertiesmust holdProperty (a) failsProperty (a) is not true not a subspace He, University of HoustonMath 2331, Linear Algebra13 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesSubspaces: Example (cont.)Another way to show thatHis not a subspace ofR2:Letu=[01]andv=[12], thenu+v=[]and sou+v=[13], which isinH. So property (b) failsand so H is not a subspace (b) failsJiwen He, University of HoustonMath 2331, Linear Algebra14 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesA Shortcut for Determining SubspacesTheorem (1)Ifv1,..,vpare in a vector spaceV, then Span{v1,..,vp}is asubspace :In order to verify this, check properties a, b and c ofdefinition of a in Span{v1.}

7 ,vp}since0=v1+v2+ +vpb. To show that Span{v1,..,vp}closed under vector addition, wechoose two arbitrary vectors in Span{v1,..,vp}:u=a1v1+a2v2+ +apvpandv=b1v1+b2v2+ + He, University of HoustonMath 2331, Linear Algebra15 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesA Shortcut for Determining subspaces (cont.)Thenu+v= (a1v1+a2v2+ +apvp) + (b1v1+b2v2+ +bpvp)= (v1+v1) + (v2+v2) + + (vp+vp)= (a1+b1)v1+ (a2+b2)v2+ + (ap+bp) +vis in Span{v1,..,vp}.c. To show that Span{v1,..,vp}closed under scalarmultiplication, choose an arbitrary numbercand an arbitraryvector in Span{v1,..,vp}:v=b1v1+b2v2+ + He, University of HoustonMath 2331, Linear Algebra16 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesA Shortcut for Determining subspaces (cont.)

8 Thencv=c(b1v1+b2v2+ +bpvp)=v1+v2+ +vpSocvis in Span{v1,..,vp}.Since properties a, b and c hold, Span{v1,..,vp}is a subspace He, University of HoustonMath 2331, Linear Algebra17 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesDetermining subspaces : RecapRecap1To show thatHis a subspace of a vector space, use show that a set is not a subspace of a vector space, providea specific example showing that at least one of the axioms a,b or c (from the definition of a subspace) is He, University of HoustonMath 2331, Linear Algebra18 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesDetermining subspaces : ExampleExampleIsV={(a+ 2b,2a 3b) :aandbare real}a subspace ofR2?Why or why not?Solution:Write vectors inVin column form:[a+ 2b2a 3b]=[a2a]+[2b 3b]=[12]+[2 3]SoV=Span{v1,v2}and thereforeVis a subspace ofbyTheorem He, University of HoustonMath 2331, Linear Algebra19 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesDetermining subspaces : ExampleExampleIsH= a+ 2ba+ 1a :aandbare real a subspace ofR3?

9 Why or why not?Solution: 0is not inHsincea=b= 0 or any other combinationof values foraandbdoes not produce the zero vector. Sopropertyfails to hold and thereforeHis not a subspace He, University of HoustonMath 2331, Linear Algebra20 / Vector Spaces & SubspacesVector Spaces subspaces Determining SubspacesDetermining subspaces : ExampleExampleIs the setHof all matrices of the form[2ab3a+b3b]a subspaceofM2 2? :Since[2ab3a+b3b]=[2a03a0]+[0bb3b]=a[ ]+b[ ].ThereforeH=Span{[2 03 0],[0 11 3]}and soHis asubspace ofM2 He, University of HoustonMath 2331, Linear Algebra21 / 21