Linear Algebra - Sebastian Pancratz

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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
Page 5 and 6: F R C F • F +
Page 7 and 8: V = C C 1, i v1, . .
Page 9 and 10: U ∪ W U ⊂ W W ⊂ U
Page 11 and 12: V W F α : V → W v, v1
Page 13 and 14: • n i=k+1 λiα(vi) = 0 α( n i
Page 15 and 16: B = {u1, . . . , un} C = {v1, . . .
Page 17: A ∈ Mm,n(F) rowrk(A) = dim rowsp
Page 20 and 21: d d(e1, . . . , en) = 1
Page 22 and 23: A A det A = 0 A A
Page 24 and 25: j det A = n i=1 (−1)i+j aij det(
Page 27 and 28: V F F R C α : V → V
Page 29 and 30: α ∈ End(V ) U V α(U) ≤ U
Page 31 and 32: deg p p l al = 0 −∞
Page 33 and 34: α p(α) = 0 mα
Page 35: n = 4 (λ−t) 4 4 4 = 3 +
Page 38 and 39: λ ∈ F θ ∈ V ∗ = α ∗ (θ
Page 41 and 42: U, V F ψ : U × V → F
Page 43: B0 = {e1, e2, . . . , en} e1 = e
Page 47: ψ : U × V → F ψL : U → V
Page 50 and 51: R n C n V = C[0, 1] 〈(, f
Page 52 and 53: = 〈αw, v〉 = 〈v, αw〉 〈v
Page 54: A n × n V n B

Linear Algebra - Sebastian Pancratz

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