線性代數(shù)大題基本思路

1.det（A）≠0時(shí)，方程組有唯一解。x1x2kx3例題：Ifthesystemoflinearequationsisxkxxk2then xx2x valueofkwillmakethesystemhasonlysolution,infiniymanysolutions,nosolutions,andwhenthesystemhasinfiniymanysolutions,finditsallsolutions. k4

A 1k2

2

①當(dāng)k23k4=0但2k28k0時(shí)k=-1不相容（notconsistent）k23k4=0但2k28k=0，即當(dāng)k=4時(shí)，方程組無窮多個(gè)解。 X4x1, is

3 ③當(dāng)k23k40k≠4且k≠-1 k

thentA） 1

X4x1,

isarbitrary。K=-1

3 2矩陣運(yùn)算，解矩陣方程.AXC=BA，C都是可逆矩陣，則等式左乘A-1，右乘C-1，則X=A-1BC- 例題：SupposeAXB=C,andA

2,B 1

C

2.Findmatrix 2 A-1= 2 0

1

00 3 23FindabasisforW,dim(W).例題：LetWbeasubspace,andletSbeaspanningsetforWFindabasisforWandcalculatedim(W)forsetS.1112S1,2,0, 2V 0VT

23 1 1 1 0 1 0 4 1 0 VTisrowequivalenttothe 0 0

1

TBT

,B

1 100 00

10So1,1,

isabasisforW, 1例題：

，findanorthogonalmatrixQsuch 1 0 QTAQ=D,whereDis hecharacteristicpolynomial

det(IA)

1Theeigenvalues

1 X 1 1 1 Theeigenvector 12=31,

1k12k2

,k1k20,k30arereal

1

1

Theeigenvector

14k4 ,k40are11 1 UsingGram-Sidtorthogonalizationprocesstothe

2,3TT 1, T

3 1Makinganorthonomalset1

1

1 2

1 1

21

3

1261 0,2 2,326

23,42

1

1

1

3 36 36orthogonal

Q

6336 63363232 53.6記住。注意點(diǎn)：注意題目是求正交化（orthogonal）還是標(biāo)準(zhǔn)正交化例題Thereisalinearindependentvectorsset{1,2,3,

0

1

Pleasegenerateanorthonormalset{u1,u2,u3}fromvectorsset{1,2,3}

1/ 1/

(3,1)

(3,2)

12112/30 (,) (,)

3 2

Generatinganor