Linear-1 4

1

A=(−4−2−4−5−32341)Find PLU decomposition of ASolution:(AII)=(−4−2−4100100−5−32010010341001001)→R3↔R1(341100001−5−32010010−4−2−4001100)→R3+43R1R2+53R1(3411000010113113−53100100103−83−4301100)→R3−1011R2(3411000010113113−531001000−6−4310111100)P=(001010100),L=(100−5310−4310111),U=(341011311300−6)

2

P=(001100010),L=(1002104−11),U=(4−32041008)Solve Ax=(123)Solution:Ax=(123)⟹PAx=P(123)=(312)⟹LUx=(312),Ux=x~⟹Lx~=(312)⟹x~=(3−5−15)⟹Ux=(3−5−15)⟹x=(141128−2532−158)
Matrix is called nilpotent, if A,A2,A3,…,Ak−1≠0,Ak=0

3a

Prove: For all nilpotent matrices A:∃B≠0:AB=0Proof:Ak=0⟹A⋅Ak−1=0,Ak−1≠0⟹∃B=Ak−1≠0:AB=0

3b

Prove: For all nilpotent matrices A:∄A−1Proof:Let ∃A−1Then Ak⋅A−1=Ak−1⋅AA−1=Ak−1I=Ak−1Ak⋅A−1=0,Ak−1≠0−Contradiction!⟹∄A−1

4a

W={A∈Rn×n|∃k∈{0}∪N:Ak=0}Prove or disprove: W is a vector subspace of vector space V=Rn×nDisproof:Let n=2Let A=(0100),B=(0010)A2=B2=0⟹A,B∈WA+B=(0110) which is an invertible matrix⟹A+B∉W⟹W is not a vector subspace of V

4b

W={A∈Rn×n|∃P∈Rn×n:P−1AP=0}Prove or disprove: W is a vector subspace of vector space V=Rn×nProof:P−1AP=0⟺PP−1APP−1=P⋅0n×n⋅P−1⟺IAI=0⟺A=0⟹W={0}⟹W is a vector subspace of V

4c

W={A∈Rn×n|AAT=I}Prove or disprove: W is a vector subspace of vector space V=Rn×nDisproof:0T=0⟹0⋅0T=0⋅0=0≠I⟹0∉W⟹W is not a vector subspace of V

4d

V={f:R→R|f is a function}W={f∈V:f(1)=f(2)}Prove or disprove: W is a vector subspace of vector space VProof:1.0V:f(x)=0⟹f(1)=0,f(2)=0⟹f(1)=f(2)⟹0V∈W2.Letf(x)=(f1+αf2)(x)=f1(x)+αf2(x)f(1)=f1(1)+αf2(1)=f1(2)+αf2(2)f(2)=f1(2)+αf2(2)=f(1)⟹(f1+αf2)(x)∈W⟹W is a vector subspace of V

4e

W={(xyz)∈R3|{x+y+z=02x+y+z=0}Prove or disprove: W is a vector subspace of V=R3Proof: 1.(000):{0+0+0=02⋅0+0+0=0⟹0∈W2.w1+αw2=(x1+αx2y1+αy2z1+αz2){x1+y1+z1⏟=0+αx2+αy2+αz2⏟=α⋅0=0=02x1+y1+z1⏟=0+2αx2+αy2+αz2⏟=α⋅0=0=0⟹w1+αw2∈W⟹W is a vector subspace of V

4f

W={A∈Rn×n|∃α∈R:AT=αA}Prove or disprove: W is a vector subspace of V=Rn×nDisproof:1.0T=0⟹∀α∈R:0T=α⋅0⟹0∈W2.X=(W1+αW2)XT=W1T+α(W2T)=α1W1+α⋅α2W2Let W1=(01−10),W2=(0110)W1T=−W1⟹W1∈WW2T=W2⟹W2∈WW1+2W2=(0310)≠(W1+2W2)T⟹(W1+2W2)∉W⟹W is not a vector subspace of V

4g

α∈R,Wα={A∈Rn×n|AT=αA}Prove or disprove: Wα is a vector subspace of V=Rn×nProof:1.0T=0⟹∀α∈R:0T=α⋅0⟹0∈W2.X=W1+αW2XT=W1T+β(W2T)=αW1+β⋅αW2=αX⟹(W1+αW2)∈W⟹W is a vector subspace of V

4h

α∈R,Wα={p(x)∈Rn[x]|p′(x)=αp(x+α2)}Prove or disprove: Wα is a vector subspace of V=Rn[x]Proof:1.p0′=0=α⋅p0(x+α2)⏟=0⟹0∈W2.p(x)=p1(x)+βp2(x)p′(x)=(p1(x)+βp2(x))′=p1′(x)+βp2′(x)=αp1(x+α2)+α⋅βp2(x+α2)==α(p(x+α2))⟹(p1(x)+βp2x)∈W⟹W is a vector subspace of V

4i

W={A∈Cn×n|A=A∗}Prove or disprove: W is a vector subspace of V=Cn×nDisproof:Let n=2Let A=(1001),B=(0−ii0)A∗=A,B∗=(0i∗(−i)∗0)=(0−ii0)=B⟹A,B∈WA+iB=(11−11)(A+iB)∗=(1−111)≠A+iB⟹A+iB∉W⟹W is not a vector subspace of V