Linear Algebra - Sebastian Pancratz

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= 〈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
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 and 44: B0 = {e1, e2, . . . , en} e1 = e
Page 45 and 46: C e1, . . . , en ψ
Page 47: ψ : U × V → F ψL : U → V
Page 50 and 51: R n C n V = C[0, 1] 〈(, f
Page 54: A n × n V n B

Linear Algebra - Sebastian Pancratz

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