Linear Algebra - Sebastian Pancratz

F R C F
• F + 0
• F \ {0} · 1
• a(b + c) = ab + ac a, b, c ∈ F
V F
V +
+ : V → V (v1, v2) ↦→ v1 + v2 ∈ V
v1, v2, v3 ∈ V (v1 + v2) + v3 = v1 + (v2 + v3)
v1, v2 ∈ V v1 + v2 = v2 + v1
0 ∈ V v + 0 = v v ∈ V
v ∈ V −v ∈ V −v + v = 0
· V
· : F × V → V (λ, v) ↦→ λv
λ ∈ F, v1, v2 ∈ V λ(v1 + v2) = λv1 + λv2
λ1, λ2 ∈ F, v ∈ V (λ1 + λ2)v = λ1v + λ2v
λ1, λ2 ∈ F, v ∈ V (λ1λ2)v = λ1(λ2v)
v ∈ V 1v = v
V F v ∈ V, λ ∈ F
0 · v = 0 λ · 0 = 0
−v = (−1) · v
λv = 0 λ = 0 v = 0
0 · v = (0 + 0) · v = 0 · v + 0 · v 0 = 0 · v
−v + v = 0 = 0v = (−1 + 1)v = (−1)v + 1v = (−1)v + v −v = (−1)v

λv = 0 λ = 0 λ −1 λ −1 (λv) = λ −1 0 = 0
λ −1 (λv) = (λ −1 λ)v = 1v = v v = 0
F n n F
X F X X → F F
V F U ⊂ V V
U ≤ V
• 0 ∈ U
• u1, u2 ∈ U =⇒ u1 + u2 ∈ U
• λ ∈ F, u ∈ U =⇒ λu ∈ U
U = ∅ U
V F U ≤ V U F
+ · V U
R R R C(R)
D(R) P (R)
n ∈ N0 λ1, . . . , λn ∈ F v1, . . . , vn ∈ V n
i=1 λivi
λ1v1 + · · · + λnvn 0
i=1 λivi = 0
S ⊂ V
v∈S λvv
v λv = 0
v1, . . . , vn V V F v ∈ V
v1, . . . , vn V = 〈v1, . . . , vn〉
S ⊂ V S V
∀v ∈ V ∃n ∈ N0 ∃v1, . . . , vn ∈ V ∃λ1, . . . , λn ∈ F v =
P2(R) 1, x, x 2
P (R)
n
λivi.
v1, . . . , vn V F
λ1v1 + · · · λnvn = 0 λ1 = · · · = λn = 0
S ⊂ V S
S
0 1 · 0 = 0
V = C R 1, i
i=1

V = C C 1, i
v1, . . . , vn V V F
P2(R) 1, x, x 2
F n e1, . . . , en ei = (0, . . . , 1, . . . , 0) T
{0} ∅
v1, . . . , vn ∈ V V F v ∈ V
v = n
i=1 λivi λi ∈ F
v ∈ V v1, . . . , vn V v = n i=1 λivi λi ∈ F
v = n i=1 µivi 0 = n i=1 (λi − µi)vi v1, . . . , vn
λi = µi i = 1, . . . , n
v ∈ V v1, . . . , vn v1, . . . , vn
V F n i=1 λivi = 0 = n i=1 0vi
λi = 0 i = 1, . . . , n
v1, . . . , vn V F {v1, . . . , vn} V
v1, . . . , vn k
α1, . . . , αk−1 ∈ F vk = α1v1 + · · · + αk−1vk−1 λ1v1 + · · · + λnvn = 0
λi = 0 k λk = 0 αi = − λi
λk
v1, . . . , vk−1, vk+1, . . . , vn V v = n i=1 λivi v = k−1 i=1 (αi +
λi)vi + n i=k+1 λivi
V
F v1, . . . , vm w1, . . . , wn V F
m ≤ n wi v1, . . . , vm, wm+1, . . . , wn V
r ≥ 0 wi wi
v1, . . . , vr, wr+1, . . . , wn V r = m r < m
vr+1 =
r
αivi +
i=1
n
i=r+1
αi, βi ∈ F βi = 0 i v1, . . . , vr, vr+1
wr+1, . . . , wn βr+1 = 0
wr+1 =
r −αi
βr+1
i=1
βiwi
vi + 1
vr+1 +
βr+1
i=r+2
−βi
wi.
βr+1
V v1, . . . , vr, vr+1, wr+2, . . . , wn m
m wi vi
m ≤ n
V F
V F dimF V

v1, . . . , vm w1, . . . , wn m ≤ n vi
wi V n ≤ m wi
vi V
dimF F n = n
dimR P2(R) = 3
dimR C = 2
dimF F = 1
V F v1, . . . , vk
k ≥ 0 v1, . . . , vk, vk+1, . . . , vn
V
v1, . . . , vk V vk+1 ∈ V \ 〈v1, . . . , vk〉
v1, . . . , vk+1 dim V − k
V U ≤ V dim U ≤ dim V
U = V
V F dim V = n
n
n
dimF V = n
v1, . . . , vn
v1, . . . , vn
v1, . . . , vn V
S ⊂ V U V
S U = 〈S〉 S U
S
U S U
V U
S
V = R R S = {1, x, x 2 , . . . } 〈S〉 = P (R)
U, W ≤ V U + W = {u + w : u ∈ U, w ∈ W } U + W ≤ V

U ∪ W U ⊂ W W ⊂ U
U W V U +W
dim U + W = dim U + dim W − dim U ∩ W
v1, . . . , vk U ∩ W v1, . . . , vk, u1, . . . , ul
U v1, . . . , vk, w1, . . . , wm W
v1, . . . , vk, u1, . . . , ul, w1, . . . , wm U + W
• v ∈ U + W v = u + w u ∈ U w ∈ W u = αivi + βiui
αi, βi ∈ F w = α ′ i vi + γiwi α ′ i , γi ∈ F
v = (αi + α ′ i)vi + βiui + γiwi.
• αivi + βiui + γiwi = 0
αivi + βiui = − γiwi
= δivi
δi ∈ F U W
U ∩ W
(αi − δi)vi + βiui = 0,
v1, . . . , vk, u1, . . . , ul U βi 0
αivi + γiwi = 0,
v1, . . . , vk, w1, . . . , wm W αi, γi 0
V F U, W ≤ V
V = U ⊕ W
v V v = u + w u ∈ U w ∈ W
W U V
U, W ≤ V V = U ⊕ W U + W = V
U ∩ W = {0}
V F U ≤ V U
V U = {0} U = V
v1, . . . , vk U u1, . . . , uk, wk+1, . . . , wn
V W = 〈wk+1, . . . , wn〉 U V
V1, . . . , Vk ≤ V Vi = { vi : vi ∈ Vi} ≤ V
Vi v ∈ V vi vi ∈ Vi
V1, . . . , Vk ≤ V
Vi

Bi Vi B = k
i=1 Bi Vi
i Vi ∩
j=i Vj = {0}
k > 2 Vi ∩ Vj = {0} i = j
=⇒ Bi Vi B = k i=1 Bi v ∈ Vi
v = k i=1 vi vi Bi
vi v B
B 0 Vi vi
Vi v1 +· · ·+vk = 0 0 vi = 0
Bi 0

V W F α : V → W
v, v1, v2 ∈ V λ ∈ F
α(v1 + v2) = α(v1) + α(v2)
α(λv) = λα(v)
D : D(R) → F (R) = R R f ↦→ f
t
x
0 : C[0, 1] → F [0, 1] f ↦→ x
0 f(t) t
A m × n α : F n → F m x ↦→ Ax
U, V, W
ιv : V → V, v ↦→ v
U β −→ V α −→ W α, β α ◦ β : U → W
V W F B V α0 : B → W
α : V → W α0 α(v) = α0(v)
v ∈ B
v ∈ V v = λ1v1 + · · · + λnvn vi ∈ B λi ∈ F
α(v) = λiα0(vi) α
V W F α : V → W
V W V → W
F
ιV : V → V
α : V → W α −1 : W → V
U α −→ V β −→ W β ◦ α : U → W

α α −1 : W → V
α −1
α −1 (w1 + w2) = α −1 (α(v1) + α(v2))
= α −1 (α(v1 + v2))
= v1 + v2
= α −1 (w1) + α −1 (w2)
α −1 (λw) = α −1 (λα(v))
= α −1 (α(λv))
= λv
= λα −1 (w)
V F n V F n
v1, . . . , vn α : V → F n , n
i=1 λivi ↦→ (λ1, . . . , λn) T
V W F
v1, . . . , vn w1, . . . , wn V W
α : V → W, λivi ↦→ λiwi
V W B
V α : V → W α(B) W
• w ∈ W w = α(v) v ∈ V v = n
i=1 λivi v1, . . . , vn ∈ B
λi ∈ F w = α( λivi) = λiα(vi)
• λ1α(v1)+· · ·+λnα(vn) = 0 α(λ1v1+· · ·+λnvn) = 0 λ1v1+· · ·+λnvn = 0
α λi = 0 i B
α : V → W ker(α) = {v ∈ V : α(v) =
0} = N(α) α Im(α) = {w ∈ W : w = α(v) v ∈ V }
N(α) ≤ V Im(α) ≤ W α N(α) = {0}
Im(α) = W n(α) = dim N(α) α
rank(α) = dim Im(α) α
V W F dimF V
α : V → W dim V = rank(α) + n(α)
v1, . . . , vk N(α) v1, . . . , vk, vk+1, . . . , vn
V α(vk+1), . . . , α(vn) Im(α)
• w ∈ Im(α) w = α(v) v ∈ V v = n i=1 λivi λi ∈ F
w = α(v) = n i=1 λiα(vi) = n k+1 λiα(vi) α(vi) = 0 i = 1, . . . , k

• n i=k+1 λiα(vi) = 0 α( n i=k+1 λivi) = 0 n i=k+1 λivi ∈ N(α)
k i=1 λivi v1, . . . , vn λi = 0
i = 1, . . . , n
V F N ≤ V V/N = {v + N : v ∈ V }
F
(v1 + N) + (v2 + N) = (v1 + v2) + N
λ(v + N) = (λv) + N
¯ V = V/N ¯v = v + N v1, . . . , vk, vk+1, . . . , vn V
v1, . . . , vk N(α) vk+1, ¯ . . . , vn ¯ ¯ V dim V/N =
dim V − dim N α : V → W W F
V/N(α) Im(α) v + N(α) ↦→ α(v) dim Im(α) =
dim V/N(α) = dim V − dim N(α)
V F α : V → V
α : V → W dim V = dim W
α
α
α
U V F L(U, V ) = {a : U → V | α }
(α1 + α1)(u) = α1(u) + α2(u)
(λα)(u) = λα(u)
u ∈ U α, α1, α2 ∈ L(U, V ) λ ∈ F F
U → V L(U, V )
U V F L(U, V ) F
U V L(U, V ) dim L(U, V ) = dim U dim V
u1, . . . , un U
v1, . . . , vm V 1 ≤ i ≤ m 1 ≤ j ≤ n εij : uk ↦→ δjkvi
1 ≤ k ≤ n εij ∈ L(U, V )
i,j λijεij = 0 1 ≤ k ≤ n
0 =
λijεij(uk) =
λikvi.
i,j
i

vi λik = 0 1 ≤ i ≤ m
1 ≤ k ≤ n εij
α ∈ L(U, V ) α(uk) =
i aikvi
(aijεij(uk)) =
aikvi = α(uk)
i,j
1 ≤ k ≤ n U
m × n F A = (aij) m n
aij ∈ F 1 ≤ i ≤ m 1 ≤ n ≤ n Mm,n(F) m × n
F
Mm,n(F)
dimF Mm,n(F) = mn
(aij) + (bij) = (aij + bij)
λ(aij) = (λaij)
1 ≤ i ≤ m 1 ≤ j ≤ n
eij = 1
Eij =
i
ei ′ j ′ = 0 (i′ , j ′ ) = (i, j).
U V F α : U → V
B = {u1, . . . , un} C = {v1, . . . , vm} U V A = (aij)
α(uj) =
i aijvi u ∈ U u =
i λiui [u]B = (λ1, . . . , λn) T
A =
[α(u1)]C · · · [α(un)]C ,
A = [α]B,C
u ∈ U [α(u)]C = [α]B,C[u]B
u ∈ U u =
j λjuj [u]B = (λ1, . . . , λn) T
α(u) =
λjα(uj)
[α(u)]C = A · [u]B
j
=
j
λj
=
i
j
i
aijvi
aijλjvi
=
(A · [u]B)ivi
i

B = {u1, . . . , un} C = {v1, . . . , vm} U, V
ε = εB : U → F n , u ↦→ [u]B φ = φC : V → F m , v ↦→ [v]C
εB
U
⏐
F n
α
−−−−→ V
⏐
⏐
φC
A·
−−−−→ F m
A [α(u)]C = A · [u]B
u ∈ U A ′ · [u]B = [α(u)]C u ∈ U
u1, . . . , un [uk]B = ek A ′ · ek k A ′
1 ≤ k ≤ n
α : U → V dim U = n dim V = m L(U, V )
Mm,n(F)
B = {u1, . . . , un} C = {v1, . . . , vm} U V
θ : L(U, V ) → Mm,n(F), α ↦→ [α]B,C
U α −→ V β −→ W B, C, D U, V, W
[β ◦ α]B,D = [β]C,D · [α]B,C
A = [α]B,C B = [β]C,D
[β ◦ α]B,D = BA
β ◦ α(uk) = β
=
j
j
ajk
=
i
j
ajkvj
i
bijwi
bijajkwi
=
(BA)ikwi
i
U V F U B = {u1, . . . , un}
B ′ = {u ′ 1 , . . . , u′ 2 } V C = {v1, . . . , vm} C ′ = {v ′ 1 , . . . , v′ m}
P = (pij) B B ′ u ′
j = i pijui
P = [u ′ 1 ]B . . . [u ′
n]B = [ιU]B ′ B.
[u]B = P [u]B ′ u ∈ U u′ j [u′ j ]B ′ = ej
P P −1 B ′ B
Q C C ′
α : U → V B, B ′ , C, C ′
A = [α]B,C A ′ = [α]B ′ ,C ′ A′ = Q −1 AP

u ∈ U
A ′ = Q −1 AP
[α(u)]C = A[u]B = AP [u]B ′
= Q[α(u)]C ′
m × n A, A ′ ∈ Mm,n(F)
Q ∈ Mm,m(F) P ∈ Mn,n(F) A ′ = QAP
Mm,n(F)
U
V m n
U V F dim U = n dim V = m
α : U →
V
r
B U C V
[α]B,C = Ir 0
0 0
m × n Ir 0
0 0
r
ur+1, . . . , un N(α) B =
{u1, . . . , ur, ur+1, . . . , un} U α(u1), . . . , α(r) Im(α)
C = {α(u1), . . . , α(ur), vr+1, . . . , vm} V
[α]B,C = Ir 0
0 0
A ∈ Mm,n(F) α : Fn → Fm , x ↦→ Ax
Fn Fm
α A A
Ir 0
0 0 r
A ∈ Mm,n(F) A rank(A)
A F n
A
α : U → V B U C V
A = [α]B,C rank(α) = rank(A)
θ : Im(α) → colsp(A), α(u) ↦→ [α(u)]C
A, A ′ ∈ Mm,n(F)
rank(A) = rank(A ′ )
A ′ = Q −1 AP Q, P α : F n → F m , x ↦→ Ax
α B, C A B ′
P C ′ Q ′ [α]B ′ ,C ′ = Q−1 AP
P C B B ′ C C ′
rank(A) = rank(α) = rank(A ′ )
A A ′
Ir 0 Ir ′ 0
r r ′
0 0 0 0
rank(A) = r rank(A ′ ) = r ′ rank(A) = rank(A ′ ) r = r ′
A A ′

A ∈ Mm,n(F) rowrk(A) = dim rowsp(A) = rank(A T )
A ∈ Mm,n(F) rowrk(A) = rank(A)
A ∈ Mm,n(F) r = rank(A) A
Ir 0
0 0
m×n
Ir 0
0 0 = QAP Q P
m×n
P T A T Q T
Ir 0
=
0 0
n×m
AT
Ir 0
0 0 n×m rank
Ir 0
0 0 = r rowrk(A) =
n×m
rank(AT ) = rank
Ir 0
0 0 = r = rank(A)
n×m
m × n
F
i j
i λ i λ ∈ F \ {0}
λ i j i = j λ ∈ F
In
Tij Mi,λ Ci,j,λ A
m × n
j 1 ij i1 ≤ i2 ≤ · · ·
ij k k < j 0
A
A n × n
In A −1
A ↦→ AE1E2 · · · Ek = I
In ↦→ InE1E2 · · · Ek = A −1 .
A n × n A
A −1 = E1 · · · Ek
A = E −1
k · · · E −1
1
n × n A A ′ A ′ = P −1 AP
P

A ∈ Mn(F) tr A = n
i=1 aii tr : Mn(F) → F
tr(AB) = tr(BA)
tr(AB) =
i j aijbji =
j i bjiaij = tr(BA)
tr(P −1 AP ) = tr(AP P −1 ) = tr(A)
α ∈ End(V ) tr(α) = tr[α]B B V
Sn {1, . . . , n}
(σ ◦ τ)(j) = σ(τ(j)) σ
+1
ε(σ) =
−1
ε : Sn → {+1, −1}
A ∈ Mn(F)
det A =
σ∈Sn
ε(σ)a σ(1)1 · · · a σ(n)n.
A (i) i A A = (A (1) , . . . , A (n) )
A n F n {e1, . . . , en}
F n
d : F n × · · · × F n → F F n
d(v1, . . . , λivi, . . . , vn) = λd(v1, . . . , vi, . . . , vn)
d(v1, . . . , vi + v ′ i, . . . , vn) = d(v1, . . . , vi, . . . , vn) + d(v1, . . . , v ′ i, . . . , vn)
i = j vi = vj d(v1, . . . , vn) = 0

d d(e1, . . . , en) = 1
i = j
d(v1, . . . , vj, . . . , vi, . . . , vn) = −d(v1, . . . , vi, . . . , vj, . . . , vn).
0 = d(v1, . . . , vi + vj, . . . , vi + vj, . . . , vn)
= 0 + d(v1, . . . , vi, . . . , vj, . . . , vn) + d(v1, . . . , vj, . . . , vi, . . . , vn) + 0
σ ∈ Sn d F n
v1, . . . , vn ∈ F n
d
d(v σ(1), . . . , v σ(n) = ε(σ)d(v1, . . . , vn)
d(e σ(1), . . . , e σ(n) = ε(σ)d(e1, . . . , en)
= ε(σ)
d F n A = (aij) = (A (1) , . . . , A (n) ) ∈ Mn(F)
d(A (1) , . . . , A (n) ) = det Ad(e1, . . . , en)
d(A (1) , . . . , A (n) ) = d(
j1
aj11ej1 , A(2) , . . . , A (n) )
=
aj11d(ej1 , A(2) , . . . , A (n) )
j1
=
aj11aj22d(ej1 , ej2 , A(3) , . . . , A (n) )
j1,j2
= · · ·
=
j1,...,jn
aj11 · · · ajnnd(ej1 , . . . , ejn)
=
aσ(1)1 · · · aσ(n)nε(σ)d(e1, . . . , en)
σ∈Sn
= (det A)d(e1, . . . , en).
d : F n × · · · × F n → F d(A (1) , . . . , A (n) ) = det A
A = (A (1) , . . . , A (n) ) d
n
j=1 a σ(j)j det A
A (k) = A (l) k = l det A = 0 τ = (kl) Sn
det A =
ε(σ)
aσ(j)j σ∈Sn
j

σ
στ ε(σ) = 1 ε(στ) = −1
det A =
⎛
⎝
aσ(j)j −
⎞
a ⎠
στ(j)j = 0
σ
det A =
σ∈Sn ε(σ) n
det A T = det A
j
a σ(1)1 · · · a σ(k)k · · · a σ(l)l · · · a σ(n)n
− a σ(1)1 · · · a σ(l)k · · · a σ(k)l · · · a σ(n)n = 0.
j=1 δ σ(j)j = ε(ι) · 1 = 1
σ ∈ Sn n
j=1 a σ(j)j = n
j=1 a jσ(j)
σ Sn σ −1 ε(σ −1 ) = ε(σ)
det A =
ε(σ)
σ∈Sn
=
ε(σ)
σ∈Sn
=
ε(σ)
σ∈Sn
= det A T
j
n
aσ(j)j j=1
n
ajσ−1 (j)
j=1
n
ajσ(j) det I
A aij = 0 i > j det A =
a11 · · · ann
det A =
σ∈Sn
j=1
ε(σ)a σ(1)1 · · · a σ(n)n.
σ(i) ≤ i i = 1, . . . , n σ(1) = 1
σ(2) = 2 σ(n) = n σ = ι det A = a11 · · · ann
E n × n A
det(AE) = det A det E = det(EA).
A det A
det Tij = −1
det Mi,λ = λ det Ci,j,λ = 1
det A −1 λ 1

A A
det A = 0
A A
det A det A = 0
A 0 A
A 0
det A = 0
A, B ∈ Mn(F) det(AB) = det(A) det(B)
A dA : (B (1) , . . . , B (n) ) ↦→ det(AB) B = (B (1) , . . . , B (n) )
F n det(AB) = dA(AB (1) , . . . , AB (n) ) dA
det(AB) = dA(B (1) , . . . , B (n) )
= det BdA(e1, . . . , en)
= det(B) det(A),
⎛
det(AB) = det ⎝
bj11A (j1)
, . . . , bjnnA (jn)
⎞
⎠
σ∈Sn
j1
=
⎛ ⎞
n
⎝ b ⎠
σ(j)j det(A σ(1) , . . . , A σ(n) )
σ∈Sn
j=1
jn
=
⎛ ⎞
n
⎝ b ⎠
σ(j)j ε(σ) det A
j=1
= det(A) det(B).
B AB det B = 0 = det(AB) B
B = E1 · · · Ek
det(AB) = det(AE1 · · · Ek)
= det A det E1 · · · det Ek
= det A det B.
A det A −1 = (det A) −1
A AA −1 = I (det A)(det A −1 ) = det I = 1
det A −1 = (det A) −1
n × n

det(P −1 AP ) = det P −1 det A det P
= (det A)(det P )(det P ) −1
= det A
α : V → V det α = det[α]B
B V
det : End(V ) → F
det ι = 1
det α ◦ β = det α det β
det α = 0 α α det α −1 =
(det α) −1
GL(V ) V GLn(F)
n × n F GL(V ) GLn(F) det : GLn(F) → F
A ∈ Mm(F) B ∈ Mk(F) C ∈ Mm,k(F) det
A C
0 B =
det A det B
B, C dB,C : A ↦→ det
A C
0 B
Fm dB,C(A) = det A det
I C
0 B C
B ↦→ det
I C
0 B Fk det I C
0 B = det B I C
0 I
det
I C
0 I = 1 I C
0 I det A C
0 B = det A det B
X =
A C
0 B
A C
det =
0 B
m+n
ε(σ) xσ(j)j σ∈§m+n j=1
x σ(j)j = 0 j ≤ m σ(j) > m σ
j ∈ [1, m] σ(j) ∈ [1, m] x σ(j)j = a σ1(j)j σ1 ∈ Sm
σ [1, m]
j ∈ [m + 1, m + k] σ(j) ∈ [m + 1, m + k] l = j − m
x σ(j)j = b σ2(l)l σ2(l) = σ(m + l) − m
ε(σ) = ε(σ1)ε(σ2) σ
det ⎛
A C
0 B = ⎝
⎞ ⎛
m
ε(σ1) a ⎠ ⎝
σ1(j)j
⎞
k
ε(σ2) a ⎠
σ2(j)j
σ1∈Sm
= det A det B
j=1
σ2∈Sk
A = (aij) n × n A ij (n − 1) × (n − 1)
A i j
l=1

j det A = n
i=1 (−1)i+j aij det(A ij )
i det A = n
j=1 (−1)i+j aij det(A ij )
det A = d(A (1) , . . . , A (n) )
=
=
n
aij(−1) i+j−2
1 ∗
d
0 A ij
i=1
n
i=1
(−1) i+j aij det A ij ,
A ∈ Mn(F) adj A n × n (i, j)
(−1) i+j det A ij
(adj A)A = (det A)I
A A−1 = 1
det A adj A
j < k
det A =
n
(adj A)jiaij = (adj A · A)jj.
i=1
0 = det(A (1) , . . . , A (k) , . . . , A (k) , . . . , A (n) )
=
n
(adj A)jiaik
i=1
= (adj A · A)jk.
1
A det A = 0
adj A
A −1 = 1
det A
det A
adj A · A = I
Ax = b m n
A m × n b F m rank A =
rank(A| b) n = rank A = rank(A| b)
x = A −1 b
m = n
A ∈ Mn(F) Ax = b x =
(x1, . . . , xn) T xi = 1
detA det A îb i = 1, . . . , n A îb
A i b

x Ax = b
det A îb = det(A(1) , . . . , A (i−1) , b, A (i+1) , . . . , A (n) )
=
j=1
= xi det A,
xi = 1
detA det Aîb i = 1, . . . , n
xj det(A (1) , . . . , A (i−1) , A (j) , A (i+1) , . . . , A (n) )
A ∈ Mn(Z) det A = ±1 b ∈ Z n Ax = b
Z

V F F
R C α : V → V
α ∈ End(V ) α B V
[α]B i = j aij = 0 α B
V [α]B i > j aij = 0
α ∈ End(V ) λ ∈ F α
v ∈ V v = 0 α(v) = λv v λ
λ α α − λι det(α − λι) = 0
λ = 0 α
χα(t) = det(α − tι) α
n = dim V A ∈ Mn(F) χα(t) = det(A − tI)
α
A λ A χA(λ) = 0 v ∈ F n
v = 0 Av = λv
χ P −1 AP (t) = det(P −1 AP − tI)
= det P −1 det(A − tI) det P
= χA(t).
1 1
0 1 2 × 2
1, 1 I
1 C
n
V C α ∈ End(V ) α
V

V C α ∈ End(V )
B V [α]B
B = {v1, . . . , vn} α(vj) ∈ 〈v1, . . . , vj〉 j = 1, . . . , n
n n = 1 n > 1
V C λ ∈ C α − λι
U = Im(α−λι) V U
α(U) ⊂ U
α(U) = α((α − λι)(V )) = (α − λι)(αV ) ≤ (α − λι)(V ) = U.
α ′ = α| U : U → U dim U < dim V
B ′ = {v1, . . . , vk} U A ′ = [α ′ ]B ′
B = {v1, . . . , vk, . . . , vn} V [α]B
[α]B =
A ′ ∗
.
0 λI
1 ≤ j ≤ k α(vj) = α ′ (vj) ∈ U k
j > k (α − λι)(vj) ∈ U U α(vj) = λvj + u u ∈ U
k j u B ′
λ (j, j)
v1 α α(v1) = λv1 U
〈v1〉 V v ∈ V v = λvv1+u u ∈ U λv ∈ F
π(v) = u V U u ∈ U ˜α : U → U ˜α(u) = π(α(u))
˜α ∈ End(U) v2, . . . , vn U ˜α(vj) ∈ 〈v2, . . . , vj〉
2 ≤ j ≤ n α(vj) = λ α(vj)vj + ˜α(vj) ∈ 〈v1, . . . , vj〉 α(v1) ∈ 〈v1〉
C
R ±I R 2
V F α ∈
End(V ) B V [α]B
χα χα F
[α]B =
⎛
⎜
⎝
a11
∗
0 ann
χα(t) = (a11 − t) · · · (ann − t) aij ∈ F
dimF V λ F U =
(α − λι)(V ) α(U) ≤ U V α ′ = α| U ∈ End(U) B ′ U
B V
[α]B =
⎞
⎟
⎠
[α ′
]B ′ ∗
.
0 λI
χα(t) = χα ′(t)χλI(t) χα ′ F
B ′ [α ′ ]B ′

α ∈ End(V ) U V α(U) ≤ U
B ′ = {v1, . . . , vk} U B = {v1, . . . , vk, . . . , vn}
V ¯ V = V/U ¯v = v + U v ∈ V ¯ B = {¯vk+1, . . . , ¯vn} ¯ V
α ′ = α| U ∈ End(U) ¯α : ¯ V → ¯ V , ¯v ↦→ ¯ α(v)
¯ V
[α ′
]B ′ ∗
[α]B =
.
0 [¯α] B¯
χα = χα ′ · χ¯α
V F α ∈ End(V ) B
V [α]B B α
α ∈ End(V ) λ1, . . . , λk α Vj =
N(α−λjι) λj V1 +· · ·+Vk Bj
Vj k j=1 Bj V1+· · ·+Vk k j=1 dim Vj = dim V
[α]B V = V1 ⊕ · · · ⊕ Vk
v1+. . .+vk = 0 vj ∈ Vj vj = 0 j = 1, . . . , k
v1 + · · · + vj = 0
α λ1
α(v1) + · · · + α(vj) − λ1v1 − · · · − λ1vj = 0
⇐⇒ (λ2 − λ1)v2 · · · + (λj − λ1)vj = 0,
Vj = Vj
V F α ∈ End(V )
F
[α]B =
⎛
⎜
⎝
λ1
0 λk
⎞
0
⎟
⎠
λ1, . . . , λk p(t) = k
j=1 (λj − t) v ∈ B α(v) = λlv
l ≤ k (λlι − α)v = 0 p(α)(v) = 0 p(α) 0
B
v ∈ V
p(t) = k
j=1 (λj − t) λ1, . . . , λk
pj(t) = (λ1 − t) · · · (λj−1 − t)(λj+1 − t) · · · (λk − t)
hj(t) = pj(t)
pj(λj)

hj(λi) = δij 1 ≤ i, j ≤ k h(t) = k
j=1 hj(t) = 1 h(t) − 1
k λ1, . . . , λk k v ∈ V
v = ι(v) = h(α)(v) =
k
hj(α)(v) =
vj = hj(α)(v) (α − λjι)vj = 0 p(α) = 0 vj
α λj v ∈ V
hj(α)
A ∈ Mn(F) P −1AP P p(A) = 0
p ∈ F[t] P
P −1 ⎛
d1
⎜
AP = D = ⎝
⎞
0
⎟
⎠
0 dn
AP = P D
AP (j) = djP (j)
j P A dj
α1, α2 ∈ End(V )
α1α2 = α2α1
B V [α1]B [α2]B
V = V1⊕· · ·⊕Vk Vj α1 α1(vj) = λjvj
vj ∈ Vj α2(Vj) ⊂ Vj v ∈ Vj α1(α2(v)) = α2(α1(v)) = α2(λjv) = λjα2(v)
α2(v) ∈ Vj α2| Vj Bj
Vj α2 α1
B α1 α2
p(t) ∈ F[t]
j=1
p(t) = ant n + · · · + a1t + a0
ai ∈ F 0 ≤ i ≤ n p(t), q(t) ∈ F[t] F[t]
m ≤ n
p(t) = ant n + · · · + a1t + a0
q(t) = bmt m + · · · + b1t + b0
(p + q)(t) = ant n + · · · + (am + bm)t m + · · · + (a1 + b1)t + (a0 + b0)
k
j=1
(pq)(t) = anbmt n+m + · · · + (a1b0 + a0b1)t + a0b0
vj

deg p p l al = 0 −∞ p
0 deg pq = deg p + deg q
F[t] a, b ∈ F[t] b = 0
q, r ∈ F[t] a = bq + r deg r < deg b r = 0 a = antn + · · · + a0
b = bmtm + · · · + b0 bm = 0 n ≥ m q = 0
r = a a a ′ = a − an
bm tn−mb deg a ′ < deg a a ′ = bq ′ + r
q ′ , r deg r < deg b q = an
bm tn−mq ′
F[t]
p ∈ F[t] λ ∈ F p(λ) = 0 q ∈ F[t] p(t) = (λ − t)q(t)
λ p e (λ − t) e p (λ − t) e+1
n n
p1, p2 n n
p, q ∈ F[t] α ∈ End(V ) p(α)q(α) = q(α)p(α)
α(v) = λv p ∈ F[t] p(α)(v) = p(λ)v
V F dim V = n
α ∈ End(V ) p n 2 p(α) = 0
dim End(V ) = n 2 a n 2, . . . , a1, a0 ∈ F
an2α n2
+ an2−1α n2−1 + · · · + a1α + a0ι = 0
n2 +1 p(t) = an2tn2 +· · ·+a1t+a0
α ∈ End(V ) mα α
mα(α) = 0
α ∈ End(V ) p ∈ F[t] p(α) = 0 mα p
F[t] p = mαq + r q, r ∈ F[t]
deg r < deg mα r = 0 p(α) = 0 = mα(α) r(α) = 0 r = 0
deg mα
V
α ∈ End(V ) χα(α) = 0
A ∈ Mn(F) χA(A) = 0
A ∈ Mn(F)
(−1) n χA(t) = t n + an−1t n−1 + · · · + a1t + a0 = det(tI − A)
B B · adj B = (det B)I adj(tI − A)
n
(tI − A)(Bn−1t n−1 + · · · + B1t + B0) = (tI − A) adj(tI − A)
= (t n + an−1t n−1 + · · · + a1t + a0)I

I = Bn−1
an−1I = Bn−2 − ABn−1
a1I = B0 − AB1
a0I = −AB0
j A n+1−j
A n + an−1A n−1 + · · · a1A + a0I = 0,
= It n + an−1It n−1 + · · · + a1It + a0I
C α ∈ End(V ) B = {v1, . . . , vn} V α(vj) ∈
〈v1, . . . , vj〉 = Uj
⎛
λ1
⎜
[α]B = ⎝
⎞
∗
⎟
⎠
0 λn
(α − λjι)Uj ⊂ Uj−i
χα(t) = (λ1 − t) · · · (λn − t)
(α − λ1ι) · · · (α − λn−1ι)(α − λnι)V
⊂(α − λ1ι) · · · (α − λn−1ι)Un−1
⊂ · · ·
mα χα
⊂(α − λ1ι)U1 = 0.
V C dim V = n
χα(t) =
k
(t − λj) aj
j=1
λ1, . . . , λk α aj λj
k
j=1 aj = n
mα(t) = k
j=1 (t − λj) ej ej 1 ≤ ej ≤ aj 1 ≤ j ≤ k
mα χα ej ≤ aj 1 ≤ j ≤ k λ α(v) = λv
v = 0 0 = mα(α)v = mα(λ)v v = 0 mα(λ) = 0 (t − λ)
mα(t)
V F α ∈ End(V )
α λ1, . . . , λk α
mα(t) = k j=1 (t − λj)

α p(α) = 0
mα
mα(t) = k j=1 (t − λj)
V a ∈ End(V )
λ1, . . . , λk α λj N(α−λjι)
λj gj = dim N(α − λjι) 1 ≤ gj ≤ aj α
aj = gj 1 ≤ j ≤ k
λj 1 ≤ gj B v1, . . . , vgj
N(α − λjι)
λjIgj [α]B =
0
∗
A ′
χα(t) = (λj − t) gj χA ′(t) gj ≤ aj
χA(t) = (−1) n t n + an−1t n−1 + · · · + a0 a0 = det A an−1 =
(−1) n−1 tr A
V C End(V )
0 1
0
J(s, λ)
⎛
λ
⎜
J(s, λ) = ⎜
⎝
1
⎞
0
⎟
1⎠
0 λ
J(s, λ) (λ − t) s
(λ − t) s λ 1
V C α ∈ End(V )
B A = [α]B
A =
⎛
⎜
⎝
B1
0 Bk
⎞
0
⎟
⎠
aj × aj Bj λj 1 ≤ j ≤ k λ = λj
Bj
Bj =
⎛
⎜
⎝
C1
0
0 Cm
⎞
⎟
⎠
s×s

m = mj Cl = J(nl, λ)
aj
aj = mj
i=1 ni gj = mj ej = n1
n1 ≥ n1 ≥ · · · nm > 0
n1 + n2 + · · · + nm = 0
C
λ1, . . . , λk
V Wj = N(α − λjι) aj V =
k j=1 Wj B = k j=1 Bj Bj Wj
⎛
B1
⎜
[α]B = ⎝
⎞
0
⎟
⎠ .
0 Bk
pj(t) = (λj −t) −aj k r=1 (λr −t) ar qj k Wj = Im(hj(α))
j=1 pjqj = 1
α(Wj) ⊂ Wj Wj α| ∈ End(Wj)
Wj
λ = λj V = Wj n = aj (α − λι) n = 0 α − λι
V
⎛
λ
⎜
⎜1
⎜
⎝
⎞
0
⎟
⎠
0 1 λ
.
λ v1, . . . , vm
α−λι v1 ↦→ v2 ↦→ . . . ↦→ vm ↦→ 0
n = 3
⎛ ⎞
⎛ ⎞ ⎛ ⎞
⎝
λ1
λ2
λ3
⎠
⎝
λ1
λ2 1 ⎠
(λ1 − t)(λ2 − t)(λ3 − t) (λ1 − t)(λ2 − t) 2
(λ1 − t)(λ2 − t)(λ3 − t) (λ1 − t)(λ2 − t) 2
⎛ ⎞
λ
⎛ ⎞
λ 1
⎝ λ ⎠
⎝ λ ⎠
λ
λ
(λ − t) 3
(λ − t) 3
λ − t (λ − t) 2
λ2
⎝
λ1
λ2
λ2
⎠
(λ1 − t)(λ2 − t) 2
(λ1 − t)(λ2 − t)
⎛ ⎞
λ 1
⎝ λ 1⎠
λ
(λ − t) 3
(λ − t) 3

n = 4 (λ−t) 4 4 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1
⎛
⎞ ⎛
⎞ ⎛
⎞
λ 1
λ 1
λ
⎟
1⎠
λ
⎜
⎝
(λ − t) 4
⎛
λ 1
⎜
⎝
λ
(λ − t) 2
λ
λ
⎞
⎟
⎠
⎜
⎝
λ 1
(λ − t) 3
⎛
λ
⎜ λ
⎝
λ − t
λ 1
λ
n((α − λι) r ) r
⎛
2 0 0
⎞
0
⎜
A = ⎜3
⎝0
2
0
0
2
−2 ⎟
0 ⎠
0 0 2 2
χA(t) = (2 − t) 4
⎛
0 0 0
⎞
0
⎜
A − 2I = ⎜3
⎝0
0
0
0
0
−2 ⎟
0 ⎠
0 0 2 0
λ
λ
λ
⎟
⎠
⎞
⎟
⎠
⎜
⎝
λ 1
λ
(λ − t) 2
⎟
λ 1⎠
λ
(A − 2I) 2 ⎛
0 0 0
⎞
0
⎜
= ⎜0
⎝0
0
0
−4
0
0 ⎟
0⎠
0 0 0 0
A mA(t) = (2 − t) 3 n(A − 2I) = 2
A
⎛
2 1 0
⎞
0
⎜
JNF = ⎜0
⎝0
2
0
1
2
0 ⎟
0⎠
0 0 0 2
.
v3 ∈ ker(A − 2I) 2 v3 = (0, 0, 1, 0) T A − 2I
⎛ ⎞
0
⎜
v3 = ⎜0
⎟
⎝1⎠
0
↦→ v2
⎛ ⎞
0
⎜
= ⎜0
⎟
⎝0⎠
2
↦→ v1
⎛ ⎞
0
⎜
= ⎜−4
⎟
⎝ 0 ⎠ ↦→ 0.
0
v4 v4 = (2, 0, 0, 3) T

V F V ∗ = L(V, F)
V V ∗ V
V F B = {e1, . . . , en} V
B ∗ = {ε1, . . . , εn} V ∗ B εj(ek) = δjk
n j=1 λjεj = 0 k = 1, . . . , n λk = ( n j=1 λjεj)(ek) = 0
B∗ ε ∈ V ∗ ε = n j=1 ε(ej)εj B∗ V ∗ ε = n
j=1 ajεj v = n
j=1 xjej
ε(v) =
n
ajxj = ⎛ ⎞
x1
⎜
a1 . . .
⎟
an ⎝ ⎠ .
j=1
F n n
U ≤ V U ◦ = {ε ∈ V ∗ : ε(u) = 0 ∀u ∈ U} U ◦ U
V ∗
U ≤ V U ◦ ≤ V ∗
U ≤ V dim U + dim U ◦ = dim V
U ≤ V e1, . . . , ek U B = {e1, . . . , ek, . . . , en}
V U ◦ = 〈εk+1, . . . , εn〉 ε1, . . . , εn V ∗ B
i > k εi(ej) = 0 j ≤ k εi ∈ U ◦ ε ∈ U ◦ ε = n
j=1 λjεj
j ≤ k λj = ε(ej) = 0 ε ∈ 〈εk+1, . . . , εn〉
U, V F U α −→ V
V ∗ α∗
−→ U ∗ α∗ (ε) = ε ◦ α ε ∈ V ∗ α
ε ◦ α : U → F α ∗ ∈ U ∗ θ1, θ2 ∈ V ∗
α ∗ (θ1 + θ2) = (θ1 + θ2) ◦ α
= θ1 ◦ α + θ2 ◦ α
xn

λ ∈ F θ ∈ V ∗
= α ∗ (θ1) + α ∗ (θ2)
α ∗ (λθ) = (λθ) ◦ α = λ(θ ◦ α) = λα ∗ (θ).
U, V F B, C
B ∗ , C ∗ U ∗ , V ∗ B, C α ∈ L(U, V )
α ∗ ∈ L(U ∗ , V ∗ ) [α ∗ ]C ∗ ,B ∗ = [α]T B,C
B = {b1, . . . , bn} C = {c1, . . . , cm} B ∗ = {β1, . . . , βn} C ∗ =
{γ1, . . . , γm} A = [α]B,C α(bj) = n
i=1 aijci
α ∗ (γr)(bs) = γr(α(bs))
n
= γr
=
=
i=1
aisci
n
aisγr(ci)
i=1
n
i=1
= ars
n
=
i=1
aisδri
ariβi
(bs)
s = 1, . . . , n α ∗ (γr) = n
i=1 ariβi [α ∗ ]C ∗ ,B ∗ = AT
det α ∗ = det α χα ∗ = χα mα ∗ = mα det A T = det A
p(A T ) = p(A) T p
U, V F α ∈ L(U, V )
α ∗ ∈ L(V ∗ , U ∗ ) ker α ∗ = (Im α) ◦ α ∗
α
ε ∈ V ∗ ε ∈ ker α ∗ α ∗ (ε) U ε ◦ α
U ε ∈ Im(α) ◦ α ∗ ker α ∗ = {0}
(Im α) ◦ = {0} Im α = V α
α ∈ L(U, V ) rank α = rank α ∗ A ∈ Mm,n(F) rank A =
rank A T
rank α ∗ = dim V ∗ − n(α ∗ )
= dim V − dim(Im α) ◦
= dim V − (dim V − dim Im α
= rank α

Im α ∗ = (ker α) ◦
V ∗ × V → F, (ε, v) ↦→ ε(v) 〈ε|v〉
U α −→ V V ∗ α∗
−→ U ∗
〈α ∗ (ε)|u〉 = 〈ε|α(u)〉
u ∈ U ε ∈ V ∗ ˆ : V → V ∗∗ , v ↦→ ˆv ˆv(ε) = ε(v)
V F ˆ : V → V ∗∗ , v ↦→ ˆv
ˆv(ε) = ε(v)
ˆv : V ∗ → F ˆ
(λ1v1 + λ2v2)(ε) = ε(λ1v1 + λ2v2) = λ1ε(v1) + λ2ε(v2)
= λ1ˆv1(ε) + λ2ˆv2(ε)
= (λ1ˆv1 + λ2ˆv2)(ε)
ε ∈ V ∗ ˆ V
dim V = dim V ∗∗ e1 = 0, e1 ∈ V e1, . . . , en V ε1, . . . , εn
V ∗ ê1(ε1) = ε1(e1) = 1 ê1 = 0
ε1, . . . , εn V ∗ E1, . . . , En V ∗∗
Ej = êj ej ∈ V ε1, . . . , εn V ∗
e1, . . . , en V
V U ≤ V V V ∗∗
U = U ◦◦ Û = U ◦◦
U ≤ U ◦◦ u ∈ U ε(u) = 0 ε ∈ U ◦ û(ε) = 0
ε ∈ U ◦ û ∈ U ◦◦ dim U = dim U ◦◦ U = U ◦◦
U1, U2 ≤ V dim V
(U1 + U2) ◦ = U ◦ 1 ∩ U ◦ 2
(U1 ∩ U2) ◦ = U ◦ 1 + U ◦ 2
V V =
P (R) V ∗ = R N
P (R) = 〈p0, p1, . . .〉 ε ∈ V ∗
(ε(p0), ε(p1), . . . )

U, V F ψ : U × V → F
u ∈ U ψ(u, v) v
v ∈ V ψ(u, v) u U = V
V
F = R V = R n ψ(x, y) = n
i=1 xiyi = x T y
V = F n A ∈ Mn(F) ψ(u, v) = u T Av
V F dim V = n B = {v1, . . . , vn}
V ψ V B A =
(ψ(vi, vj)) = [ψ]B
ψ V B V ψ(u, v) =
[u] T B [ψ]B[v]B u, v ∈ V
B = {v1, . . . , vn} u = n
i=1 aivi v = n
i=1 bivi
ψ(u, v) = ψ( aivi, bjvj)
=
aibjψ(vi, vj)
i,j
⎛
b1
bn
⎞
= ⎜
a1 · · ·
⎟
an [ψ]B ⎝ ⎠
= [u] T B[ψ]B[v]B.
[ψ]B u, v ∈ V ψ(u, v) =
[u] T B A[v]B u, v ∈ V u = vi v = vj Aij = ψ(vi, vj)
B = {v1, . . . , vn}, B ′ = {v ′ 1 , . . . , v′ n} V P
B B ′
v ′ j =
n
i=1
pijvi
[ψ]B ′ = P T [ψ]BP
[v]B = P [v]B ′

u, v ∈ V
ψ(u, v) = [u] T B[ψ]B[v]B
[ψ]B ′ = P T [ψ]BP [ψ]B ′
= (P [u]B ′)T [ψ]B(P [v]B ′)
= [u] T B ′P T [ψ]BP [v]B ′,
A, B B = P T AP
P
Mn(R)
ψ ψ(u, v) = ψ(v, u) u, v ∈ V
A = [ψ]B A = A T
P T AP = D P A = A T
V Q : V → R
V Q(λv) = λ 2 Q(v) λ ∈ R, v ∈ V
ψ V Q(v) + Q(w) + 2ψ(v, w) = Q(v + w) v, w ∈ V
ψ Q(v) = ψ(v, v)
(Q(v + w) − Q(v) − Q(w))
Q ψ(v, w) = 1
2
⎛
Ip
⎝ −Iq
⎞
0
⎠ .
0 0
dim V = n
ψ(u, v) = 0 u, v ∈ V [ψ]B = 0
B e ∈ V ψ(e, e) = 0
2ψ(u, v) = ψ(u + v, u + v) − ψ(u, u) − ψ(v, v) 0 u, v ∈ V
W = {v ∈ V : ψ(e, v) = 0} V = 〈e〉 ⊕ W v ∈ V v = λe + (v − λe)
〈e〉 ∩ W = {0}
λ ∈ R λ v − λe ∈ W λ = ψ(e,v)
ψ(e,e)
ψ(e, λe) = 0 λ = 0 ψ ′ ψ W
e2, . . . , en W ψ ′
ψ ′
⎛
⎜
⎝
d2
0 dn
⎞
0
⎟
⎠ .

B0 = {e1, e2, . . . , en} e1 = e [ψ]B0
⎛
d1
⎜ 0
⎜
⎝
0
d2
⎞
0
⎟
⎠
0 dn
,
d1 = ψ(e, e)
B0 d1, . . . , dp > 0 dp+1, . . . , dp+q < 0 di = 0
1
i > p + q 1 ≤ i ≤ p + q ei √|di| ei B
rank ψ = p + q
⎛
Ip
[ψ]B = ⎝ −Iq
⎞
0
⎠ .
0 0
ψ s(ψ) = p − q
p = 1
2
1
(r + s) q = 2 (r − s) (p, q)
ψ
⎛
Ip
⎝ −Iq
⎞
0
⎠
⎛
Ip ′
⎝ −Iq
0
0 0
′
⎞
⎠
0 0
p = p ′ q = q ′
B = {v1, . . . , vp, vp+1, . . . , vp+q, vp+q+1, . . . , vn}
⎛
Ip
[ψ]B = ⎝ −Iq
⎞
0
⎠ .
0 0
X = 〈v1, . . . , vp〉 Y = 〈vp+1, . . . , vn〉 ψ X
p V ψ
Q( p i=1 λivi) = p i=1 λ2i ≥ 0 p i=1 λivi = 0 ψ
X X ′ ≤ V dim X ′ = p ′ ψ X ′
X ′ ∩ Y = {0} ψ Y dim X ′ + dim Y ≤ n
p ′ = dim X ′ ≤ n − dim Y = p
N = 〈vp+1, . . . , vp+q〉 ψ N q
V ψ
p q
p ψ X
p ψ q N

t = min{p, q} ψ 〈v1 +
vp+1, . . . , vt + vp+t, vp+q+1, . . . , vn〉 0 n − max{p, q}
ψ 0 ψ = 0 U ≤ V U ∩ X = {0} = U ∩ N
dim U ≤ n − p dim U ≤ n − q
ψ {v ∈ V : ψ(v, w) = 0 ∀w ∈ V } ker ψ =
〈vp+q+1, . . . , vn〉
ψ ker ψ = {0}
[ψ]B B n = p + q
Q V = R 3
Q(x1, x2, x3) = x 2 1 + x 2 2 + 2x 2 3 + 2x1x2 + 2x1x3 − 2x2x3.
Q
⎛
1 1
⎞
1
A = ⎝1
1 −1⎠
.
1 −1 2
Q(x1, x2, x3) = x 2 1 + x 2 2 + 2x 2 3 + 2x1x2 + 2x1x3 − 2x2x3
rank(Q) = 3 s(Q) = 2 − 1 = 1
= (x1 + x2 + x3) 2 + x 2 3 − 4x2x3
= (x1 + x2 + x3) 2 + (x3 − 2x2) 2 − (2x2) 2
P −1 ⎛
1 1
⎞
1
= ⎝0
−2 1⎠
P
0 2 0
T ⎛
1 0
⎞
0
AP = ⎝0
1 0 ⎠
0 0 −1
P P T AP
P = E1 · · · Ek
A → E T 1 AE1 → . . . → E T k · · · ET 1 AE1 · · · Ek = D
e1 Q(e1) = 0 e1 = (1, 0, 0) T
Q(e1) = 1 W = {v ∈ V : ψ(e, v) = 0} = {(a, b, c) T : a + b + c = 0}
e T 1 A = (1, 1, 1) e2 ∈ W Q(e2) = 0 e2 = (1, 0, −1) Q(e2) = 1
e3 ∈ W ψ(e2, e3) = 0 e3 = (a, b, c) T a + b + c = 0
2b − c = 0 e T 2 A = (0, 2, −1) e3 = 1
2 (−3, 1, 2)T Q(e3) = −1
s(Q)

C
e1, . . . , en ψ
⎛ ⎞
d1 0
⎜
⎝
⎟
⎠ .
0 dn
d1, . . . , dr = 0 di = 0 i > r
ej 1 √ ej 1 ≤ j ≤ r ψ
dj
Ir 0
.
0 0
P T AP =
Ir 0
0 0
P rank A = r
V V
ψ : V × V → C
u ∈ V v ↦→ ψ(u, v)
u, v ∈ V ψ(u, v) =
¯
ψ(v, u)
ψ
ψ(u, v) =
ψ(u, λ1v1 + λ2v2) = λ1ψ(u, v1) + λ2ψ(u, v2)
ψ(λ1u1 + λ2u2, v) = ¯ λ1ψ(u1, v) + ¯ λ2ψ(u2, v)
¯
ψ(v, u)
ψ V
Q : V → C Q(v) = ψ(v, v) Q(v) ∈ R v ∈ V
Q(λv) = |v| 2 Q(v) ψ
ψ(u, v) = 1
4 (Q(u + v) − Q(u − v) − iQ(u + iv) + iQ(u − iv))
B = {v1, . . . , vn} V ψ B [ψ]B =
(ψ(vi, vj)) = A ψ(u, v) = [ψ] T
B [ψ]B[v]B A = ĀT
¯ P T AP
P
ψ V
B V ψ
⎛
Ip
⎝ −Iq
⎞
0
⎠
0 0
p q ψ B
v1, . . . , vn
n
Q( ξivi) = |ξ1| 2 + · · · + |ξp| 2 − |ξp+1| 2 − · · · − |ξp+q| 2 .
i=1

ψ V p = n
ψ(u, v) = −ψ(v, u)
u, v ∈ V ψ(u, u) = 0 u ∈ V A = [ψ]B B V
A T = −A A
A
A = 1
2 (A + AT ) + 1
2 (A − AT )
ψ V
v1, w1, . . . , vm, wm, v2m+1, . . . , vn ψ
⎛
0 1
⎞
0
⎜
⎜−1
⎜
⎝
0
0
−1
1
0
0
⎟ .
⎟
⎠
0 0
dim V = n ψ = 0
v1, w1 ψ(v1, w1) = 1 ψ(w1, v1) = −1 U = 〈v1, w1〉
W = {v ∈ V : ψ(v1, v) = 0 = ψ(w1, v)} V = U ⊕ W v = (av1 + bw1) +
(v − av1 − bw1) a = ψ(v, w1), b = (ψ(v1, v) v − av1 − bw1 ∈ W
U ∩ W = {0} av1 + bw1 ∈ W ψ(av1 + bw1, av1 − bw1) = a 2 + b 2
W
ψ n = 2m
B v1, . . . , vm, w1, . . . , wm, v2m+1, . . . , vn
⎛
0
⎝−Im
Im
0
⎞
0
0⎠
.
0 0 0
U × V U, V
F ψ : U × V → F
U = V ∗ ψ : V ∗ × V → F, (α, v) ↦→ α(v)
F = C ¯ V V V
· λ · v = ¯ λv
¯V × V

ψ : U × V → F
ψL : U → V ∗ , u ↦→ (ψL(u) : v ↦→ ψ(u, v))
ψR : V → U ∗ , v ↦→ (ψR(v) : u ↦→ ψ(u, v))
ψ ker ψL = {0} ker ψR = {0}
ψ dim U = dim V dim U ≤ V ∗ = dim V dim V ≤
dim U ∗ = dim U
dim U = dim V ker ψL = {0} ker ψR = {0}
ψ U × V u1, . . . , un U
ψL(u1), . . . , ψL(un) V ∗ v1, . . . , vn V
ψ(ui, vj) = δij
ψ V W ≤ V W ⊥ =
{v ∈ V : ψ(w, v) = 0 ∀w ∈ W } W ⊥ ≤ V dim W + dim W ⊥ = dim V
u1, . . . , un V u1, . . . , um W
v1, . . . , vn W ⊥ = 〈vm+1, . . . , vn〉
ψL : V → V ∗ , u ↦→ (ψL(u) : V → F, v ↦→ ψ(u, v)).
ψ W ⊥ = (ψL(W )) ◦
v ∈ W ⊥ ⇐⇒ ψ(w, v) = 0 ∀w ∈ W
⇐⇒ ψL(w)(v) = 0 ∀w ∈ W
⇐⇒ v ∈ (ψL(W )) ◦ .
dim W + dim W ⊥ = dim W + dim(ψL(W )) ◦
= dim W + dim V − dim ψL(W )
= dim V.

V V
V 〈v, w〉
(v, w) ∈ V × V V
〈, 〉 〈v, v〉 > 0
v ∈ V \ {0}
v v = 〈v, v〉 |v| > 0 v = 0
v, w ∈ V |〈v, w〉| ≤ vw
v = 0 v = 0
• t ∈ R
t = 〈v,w〉
v 2
• t ∈ C
0 ≤ tv − w 2 = t 2 v 2 − 2t〈v, w〉 + w 2 .
0 ≤ tv − w 2 = t¯tv 2 − ¯t〈v, w〉 − t〈v, w〉 + w 2 .
t = 〈v,w〉
v 2 ¯t = 〈v,w〉
v 2
v, w = 0 θ
θ ∈ [0, 2π)
cos θ = 〈v,w〉
vw
v, w ∈ V v + w ≤ v + w
v + w 2 = v 2 + 〈v, w〉 + 〈v, w〉 + w 2
≤ v 2 + 2vw + w 2
= (v + w) 2
d(v, w) = v − w V

R n C n
V = C[0, 1] 〈(, f〉, g) = 1
0 f(t)g(t) dt
{e1, . . . , ek} 〈ei, ej〉 = 0 i = j
ej = 1 j
v = k
j=1 λjej
λj = 〈ej, v〉
v1, . . . , vn V
e1, . . . , en V 〈v1, . . . , vk〉 = 〈e1, . . . , ek〉
1 ≤ k ≤ n
e1 = 1
v1 v1 e1, . . . , ek e ′ k+1 = vk+1 −
k
j=1 λjej λj 〈ej, e ′ k+1 〉 = 0 1 ≤ j ≤ k λj = 〈ej, vk+1〉
e ′ k+1 = 0 v1, . . . , vk, vk+1 ek+1 = 1
e ′ k+1 e′ k+1
〈ej, ek+1〉 = 0 1 ≤ j ≤ k ek+1 = 1 〈e1, . . . , ek+1〉 = 〈v1, . . . , vk+1〉
λj = 〈ej,vk+1〉
〈ej,ej〉
e1, . . . , ek
e1, . . . , ek, vk+1, . . . , vn V
e1, . . . , ek, ek+1, . . . , en V
A A = RT
R T
A A = UT U
T
R n v1, . . . , vn
A (1) , . . . , A (n) A e1, . . . , en
R R (j) = ej
R T R = I vk = n
j=1 tjkej vk ∈ 〈e1, . . . , ek〉
T = (tij) A = RT A (k) = n
j=1 tjkR ( j)
A C n
R U Ū T U = I U
V W ≤ V W ⊥ = {v ∈ V : v ⊥
w ∀w ∈ W } W V
V W ≤ V
V = W ⊕ W ⊥

e1, . . . , ek W e1, . . . , ek, ek+1, . . . , en
V ek+1, . . . , en W ⊥
v ∈ V v = k j=1 λjej + n j=k+1 λjej V = W + W ⊥ W ∩ W ⊥ = {0}
〈v, v〉 = 0 v = 0
V W ≤ V
π = πW V W π 2 = π W = Im π W ⊥ = ker π
e1, . . . , ek W
e1, . . . , ek, ek+1, . . . , en V v = n j=1 λjej πw(v) =
k j=1 λjej πw(v) = k j=1 〈ej, v〉ej
ιV = πW + π W ⊥ πW π W ⊥ = 0
W ≤ V v ∈ V πW (v) W v
w0 = πW (v) d(w0, v) ≤ d(w, v) w ∈ W
V α ∈ End(V )
α ∗ V α
α ∗ : V → V 〈αv, w〉 = 〈v, α ∗ w〉
v, w ∈ V B V [α ∗ ]B = [α] T
B
B = {e1, . . . , en} V A = [α]B = (aij)
α ∗ [α ∗ ]B = ĀT C = ĀT 1 ≤ i, j ≤ n
〈αei, ej〉 = 〈
=
n
k=1
akiek, ej〉
n
aki〈ek, ¯ ej〉
k=1
= ¯
aji
= cij
n
= ckj〈ei, ek〉
k=1
= 〈ei,
n
k=1
= 〈ei, α ∗ ej〉
ckjek〉
〈αv, w〉 = 〈v, α ∗ w〉 v, w ∈ V α ∗
[α ∗ ]B = ĀT
(α + β) ∗ = α ∗ + β ∗ (λα) ∗ = ¯ λα ∗
α ∗∗ = α ι ∗ = ι (αβ) ∗ = β ∗ α ∗
〈α ∗ v, w〉 = 〈v, α ∗∗ w〉
〈α ∗ v, w〉 = 〈w, α ∗ v〉

= 〈αw, v〉
= 〈v, αw〉
〈v, αw − α ∗∗ w〉 = 0 v ∈ V αw = α ∗∗ w w ∈ V α ∗∗ = α
A n × n
• A A T = A ĀT = A A
• A A T = A −1 ĀT = A −1 A
V R C α ∈ End(V )
• α α = α ∗ 〈αv, w〉 = 〈v, αw〉
v, w ∈ V
• α α ∗ = α −1 〈αv, αw〉 = 〈v, w〉 α
V
• α αα ∗ = α ∗ α
V α ∈ End(V ) B
V α
[α]B
V α ∈
End(V ) α
V α
V C λ ∈ C e ∈ V
α(e) = λe e = 1 W = 〈e〉 ⊥ W α w ∈ W
〈α(w), e〉 = 〈w, α ∗ (e)〉 = 〈w, α(e)〉 = 〈w, λe〉 = λ〈w, e〉 = 0
α(w) ∈ W w ∈ W
〈α(w), e〉 = 〈w, α ∗ (e)〉 = 〈w, α −1 (e)〉 = 〈w, λ −1 e〉 = λ −1 〈w, e〉 = 0
α(w) ∈ W V = 〈e〉 ⊕ W α| W
W e2, . . . , en W
α B = {e1, e2, . . . , en}
α
[α]B = [α] T
B [α]B
[α]B =
⎛
⎜
⎝
λ1
0 λn
⎞
0
⎟
⎠ ,
¯ λj = λj 1 ≤ j ≤ n λj [α−1 ]B = [α] T
B ¯ λj = λ −1
j
1 ≤ j ≤ n |λj| = 1

α ∈ End(V ) v1, v2 α
λ1, λ2 v1 ⊥ v2
〈α(v1), v2〉
¯λ1〈v1, v2〉 = 〈α(v1), v2〉 = v1α(v2) = λ2〈v1, v2〉
λ1 = ¯ λ1 = λ2 〈v1, v2〉 = 0 λ1 = λ2
α α
B V [α]B
[α]B α
V α ∈ End(V )
V α
V α ∈ End(V )
α
B V
⎛
1 0
⎞
⎜0
1
⎟
θi ∈ R
⎜
[α]B = ⎜
⎝
−1 0
0 −1
cos θ1 sin θ1
− sin θ1 cos θ1
⎟
⎠
A
R n C n
v1, . . . , vn A
P = (v1, . . . , vn) AP = P D D
P −1 AP = D = P T AP ¯ P T AP = D
ψ
V A B
V ψ
A
A
P P −1 AP = D = P T AP
¯P T AP = D

A n × n V
n B α ∈ End(V ) [α]B = A
ψ [ψ]B = A C
A 〈, 〉 V B
〈ei, ej〉 = δij C 〈, 〉 P
P −1 = P T P −1 AP = P T AP [α]C = [ψ]C
ψ φ
V
ψ V ψ
φ
A, C ψ, φ
ψ P P T AP = I ψ P T CP
Q QT P T CP Q
QT P T AP Q = QT IQ = I
Q T P T CP Q = D =
⎛
⎜
⎝
d1
0 dn
⎞
0
⎟
⎠ .
d1, . . . , dn D det(C − tA)
det(D − tI)
det(D − tI) = det((P Q) 2 (C − tA)(P Q)) = (det(P Q)) 2 det(C − tA),
det(D − tI) det(C − tA)
V
α ∈ End(V ) α ∗ ∈ End(V ) 〈α(v), w〉 = 〈v, α ∗ (w)〉
w ∈ V φ(w) : V → F, v ↦→ 〈v, w〉 V
¯V → V ∗ , w ↦→ φ(w) ¯ V V λ · v = ¯ λv
V w ′ ∈ V v ↦→ 〈v, w ′ 〉
w ∈ V v ↦→ 〈α(v), w〉 V w ′ = α ∗ (w)
〈α(v), w〉 = 〈v, α ∗ (w)〉 v, w ∈ V α ∗