LINEAR ALGEBRA FINAL EXAM
algebraicoomultiplicityoo-ooans--
✔✔theoomultiplicityooofooanooeignevalueooasooaoorootooofootheoocharacteristicooequation
basicoovarriableoo-ooans--
✔✔aoovariableooinooaoolinearoosystemoothatoocorrespondsootoooaoopivotoocolumnooinootheooco
efficientoomatrix
basisoo-ooans--
✔✔anooindexedoosetooBoo=oo{v1,...,vP}ooinooVoosuchoothatoo(i)ooBooisooaoolinearlyooindepende
ntoosetooandoo(ii)ootheoosubspaceoospannedoobyooBoocoincidesoowithooH,oothatoois,ooHoo=ooSp
an{v1,...,vP}oo
(maximaloolinearlyooindependentooset)
bestooapproximationoo-ooans--
✔✔theooclosestoopointooinooaoogivenoosubspaceootoooaoogivenoovector
characteristicooequationoo-ooans--✔✔det(A-λI)oo=oo0
codomainoo(ofooaootransformationooT)oo-ooans--
✔✔theoosetooR^moothatoocontainsootheoorangeooofooT.ooInoogeneral,ooifooToomapsooaoovectoro
ospaceooVoointoooaoovectoroospaceooW,oothenooWooisoocalledootheoocodomainooofooT
columnoospaceoo-ooans--
✔✔theoosetooColooAooofooalloolinearoocombinationsooofootheoocolumnsooofooA.ooIfooAoo=oo{a1...
aN}oothenooColooAoo=ooSpanoo{a1,...,aN}
consistentoolinearoosystemoo-ooans--✔✔aoolinearoosystemoowithooatooleastoooneoosolution
diagonalizableoo-ooans--
✔✔aoomatrixoothatoocanoobeoowrittenooinoofactoredooformooasooPDP⁻¹,oowhereooDooisooaoodia
gonaloomatrixooandooPooisooanooinvertibleoomatrix
diagonaloomatrixoo-ooans--
✔✔aoosquareoomatrixoowhoseooentriesoonotooonootheoomainoodiagonalooareooalloozero
dimensionooofooaoosubspaceooSoo-ooans--✔✔theoonumberooofoovectorsooinooaoobasisooforooS
, domainoo(ofooaootransformationooT)oo-ooans--
✔✔theoosetooofooalloovectorsooxooforoowhichooT(x)ooisoodefined
eigenspaceoo-ooans--
✔✔theoosetooofooalloosolutionsooofooAxoo=ooλx,oowhereooλooisooanooeigenvalueooofooA.ooConsi
stsooofootheoozerooovectorooandooallooeigenvectorsoocorrespondingootoooλ
eigenvalueoo-ooans--
✔✔aooscalarooλoosuchoothatootheooequationooAxoo=ooλxoohasooaoosolutionooforoosomeoononze
rooovectoroox
eigenvectoroo-ooans--
✔✔aoononzerooovectorooxoosuchoothatooAxoo=ooλxooforoosomeooscalarooλ
Gram-Schmidtooprocessoo-ooans--
✔✔anooalgorithmooforooproducingooanooorthogonalooorooorthonormaloobasisooforooaoosubspa
ceoothatooisoospannedoobyooaoogivenoosetooofoovectors
Homogeneousooequationoo-ooans--✔✔Axoo=oo0
imageoo(ofooaoovectorooxoounderooaootransformationooT)oo-ooans--
✔✔TheoovectorooT(x)ooassignedootoooxoobyooT
inconsistentoo-ooans--✔✔aoolinearoosystemoowithoonooosolution
indefiniteooquadraticooformoo-ooans--
✔✔aooquadraticooformooQoosuchoothatooQ(x)ooassumesoobothoopositiveooandoonegativeooval
ues
infinite-dimensionaloo-ooans--
✔✔aoononzerooovectoroospaceooVoothatoohasoonooofiniteoobasis
inverseoo(ofooaoomatrix)oo-ooans--✔✔AA⁻¹oo=ooA⁻¹Aoo=ooI
invertibleoolinearootransformationoo-ooans--
✔✔aoolinearootransformationooT:ooR^Noo→R^NoosuchoothatoothereooexistsooaoofunctionooS:oo
R^Noo→R^NoosatisfyingoobothooT(S(x))oo=ooxooandooS(T(x))oo=ooxooforooallooxooinooR^n
invertibleoomatrixoo-ooans--✔✔aoosquareoomatrixoothatoopossessesooanooinverse
isomorphicoovectoroospacesoo-ooans--
✔✔twooovectoroospacesooVooandooWooforoowhichoothereooisooaooone-to-
oneoolinearootransformationooToothatoomapsooVooontoooW
isomorphismoo-ooans--✔✔aooone-to-
oneoolinearoomappingoofromoooneoovectoroospaceootoooanother
algebraicoomultiplicityoo-ooans--
✔✔theoomultiplicityooofooanooeignevalueooasooaoorootooofootheoocharacteristicooequation
basicoovarriableoo-ooans--
✔✔aoovariableooinooaoolinearoosystemoothatoocorrespondsootoooaoopivotoocolumnooinootheooco
efficientoomatrix
basisoo-ooans--
✔✔anooindexedoosetooBoo=oo{v1,...,vP}ooinooVoosuchoothatoo(i)ooBooisooaoolinearlyooindepende
ntoosetooandoo(ii)ootheoosubspaceoospannedoobyooBoocoincidesoowithooH,oothatoois,ooHoo=ooSp
an{v1,...,vP}oo
(maximaloolinearlyooindependentooset)
bestooapproximationoo-ooans--
✔✔theooclosestoopointooinooaoogivenoosubspaceootoooaoogivenoovector
characteristicooequationoo-ooans--✔✔det(A-λI)oo=oo0
codomainoo(ofooaootransformationooT)oo-ooans--
✔✔theoosetooR^moothatoocontainsootheoorangeooofooT.ooInoogeneral,ooifooToomapsooaoovectoro
ospaceooVoointoooaoovectoroospaceooW,oothenooWooisoocalledootheoocodomainooofooT
columnoospaceoo-ooans--
✔✔theoosetooColooAooofooalloolinearoocombinationsooofootheoocolumnsooofooA.ooIfooAoo=oo{a1...
aN}oothenooColooAoo=ooSpanoo{a1,...,aN}
consistentoolinearoosystemoo-ooans--✔✔aoolinearoosystemoowithooatooleastoooneoosolution
diagonalizableoo-ooans--
✔✔aoomatrixoothatoocanoobeoowrittenooinoofactoredooformooasooPDP⁻¹,oowhereooDooisooaoodia
gonaloomatrixooandooPooisooanooinvertibleoomatrix
diagonaloomatrixoo-ooans--
✔✔aoosquareoomatrixoowhoseooentriesoonotooonootheoomainoodiagonalooareooalloozero
dimensionooofooaoosubspaceooSoo-ooans--✔✔theoonumberooofoovectorsooinooaoobasisooforooS
, domainoo(ofooaootransformationooT)oo-ooans--
✔✔theoosetooofooalloovectorsooxooforoowhichooT(x)ooisoodefined
eigenspaceoo-ooans--
✔✔theoosetooofooalloosolutionsooofooAxoo=ooλx,oowhereooλooisooanooeigenvalueooofooA.ooConsi
stsooofootheoozerooovectorooandooallooeigenvectorsoocorrespondingootoooλ
eigenvalueoo-ooans--
✔✔aooscalarooλoosuchoothatootheooequationooAxoo=ooλxoohasooaoosolutionooforoosomeoononze
rooovectoroox
eigenvectoroo-ooans--
✔✔aoononzerooovectorooxoosuchoothatooAxoo=ooλxooforoosomeooscalarooλ
Gram-Schmidtooprocessoo-ooans--
✔✔anooalgorithmooforooproducingooanooorthogonalooorooorthonormaloobasisooforooaoosubspa
ceoothatooisoospannedoobyooaoogivenoosetooofoovectors
Homogeneousooequationoo-ooans--✔✔Axoo=oo0
imageoo(ofooaoovectorooxoounderooaootransformationooT)oo-ooans--
✔✔TheoovectorooT(x)ooassignedootoooxoobyooT
inconsistentoo-ooans--✔✔aoolinearoosystemoowithoonooosolution
indefiniteooquadraticooformoo-ooans--
✔✔aooquadraticooformooQoosuchoothatooQ(x)ooassumesoobothoopositiveooandoonegativeooval
ues
infinite-dimensionaloo-ooans--
✔✔aoononzerooovectoroospaceooVoothatoohasoonooofiniteoobasis
inverseoo(ofooaoomatrix)oo-ooans--✔✔AA⁻¹oo=ooA⁻¹Aoo=ooI
invertibleoolinearootransformationoo-ooans--
✔✔aoolinearootransformationooT:ooR^Noo→R^NoosuchoothatoothereooexistsooaoofunctionooS:oo
R^Noo→R^NoosatisfyingoobothooT(S(x))oo=ooxooandooS(T(x))oo=ooxooforooallooxooinooR^n
invertibleoomatrixoo-ooans--✔✔aoosquareoomatrixoothatoopossessesooanooinverse
isomorphicoovectoroospacesoo-ooans--
✔✔twooovectoroospacesooVooandooWooforoowhichoothereooisooaooone-to-
oneoolinearootransformationooToothatoomapsooVooontoooW
isomorphismoo-ooans--✔✔aooone-to-
oneoolinearoomappingoofromoooneoovectoroospaceootoooanother
