Esercizi svolti Algebra lineare

RIPASSO PARTE 2

ES 1.1.1

z = 1 + i√2

1/z = z/|z|² = (1 - i√2) / (1 + i²) = (1 - i√2) / (1 - 2) = -1 + i√2

ES 1.2.1

x² + x = 0

x(x + 1) = 0

S = {0, 1}

ES 1.2.2

x² + x + 1 = 0

S = ∅

ES 1.2.3

x² = 1

S = {1}

ES 1.2.4

|z| = |ẑ|

|x + iy| = |x - iy|

√(x² + (i²y)²) = √(x² + (-i)²y²)

√(x² - y²) = √(x² - y²)

t ₍ ¹⁄_{2 )} + ( -1 ₍ ¹⁄_{3 )} = ( t-1, t+1, 2t+3 )

ES 488

a) ₍ ¹⁄_{1 )}

ES 481

p(λ) = 1-λ 1 0 0 2-λ 1 0 0 3-λ = (1-λ)(2-λ)(3-λ) = 0

S = { 1, 2, 3 }

ES 468

λ₁ + λ₂ = 0

0 + 0 = 1

ES 144

Co(A)^T =

A^-1 =

Es 29.2.13

Π: x+y+z=2 x+y+z-2=0

r=span ¹ _{1 1} ¹ _{1 0}

ES 529

• x=0
• z+y=0

¹ _{0 0} ⁰ _{1 1} ⁰

ES 554

(x+y+z=0

y=-t

(-1, -1)

(0, 0)

¹ _{1 0} ¹ ₀

¹ _{0 1} ⁰ _{0 0 0}

ES 757

b(p, q) = p(0)q(0)

b = | b(1, 1) b(1, x) b(1, x²) | | b(x, 1) b(x, x) b(x, x²) | | b(x², 1) b(x², x) b(x², x²) | = | 1 0 0 | | 0 0 0 | | 0 0 0 |

ρ(x) = | 1-x 0 0 | | 0 -x 0 | | 0 0 -x | = (-x)²(1-x)

x²(1-x) = 0

S = {0,0,1}

mg(0) = dim(Ker A) = dim(V) - R(0) = 3 - 1 = 2

ES 770

:(

ES 772

1. | 1 1 | | 1 1 | = 1 - 1 = 0
2. | 0 1 | | 1 0 | = 0 - 1 = - 1
3. | 1 1 | | 1 2 | = 2 - 1 = 1
4. | 1 2 | | 2 2 | = 2 - 4 = - 2

ES 64

f(x, y) = (4x - 4y, 4x - 4y)

A = 2 ^{4 -4} 4 4 ^{4 -4} t = 16 ^{-4 4} 4x 4x ^x t ² + 16 x ² = 0 S = { 0 , 0 } rk(A) = dim (V) - R(A) = 1

ES 160

r = { (x, y, z) | x - y = y - z = 1 } (x) (y) (z) t = ³ + t ^T

x - y = 1

y - z = 1

x - y = y - z

x = 2y - z

s = span (1, 2, 1) S = t (1)

{ x = t y = 2t z = t }

{ x - y = 1 x - 2x = 1 x - x = 1

NO

ES 378

P(4, 0, -1)

r(t): (t, 4t+1, -2t-1)

r(t); t

_{(1) (4) (-2)}

P-x₀ = _{(0) (4) (-1) (1)}

d (P, r) = = _{(4) (-1) (0)}

= √(4² + 1²) = √17

ES 450

{ x-y+z=0

{ x+y+z≥0

x-y = x+y

X=1

Y=1

Y=0

Z=0

Z=1

X=0

Y=1

Z=0

Y=0

Z=1

ES 843

(1, i, i)

B:

• λ₂ = i
• λ₁ + λ₂ + iλ₃ = i
• λ₁ = 1

1 + 1 + iλ₃ = i

2 + iλ₃ = i

• iλ₃ = i - 2
• -λ₃ = -1 - 2i
• λ₃ = 1 + 2i

ES 842

X = { x + y - 4z + 1 = 0 }

x + y - 4z + 1 = 0

x = -y + 4z - 1

ES 858

f(x) = ^{( 1 1)} _{( 0 1)} x Macio di Jordan

ES 842

X = { x+y-4z+1=0 }

X = r ⁽¹⁾ _{(0) (0)} + s ⁽⁰⁾ _{(1) (0)} + t ⁽⁰⁾ _{(0) (-4)}

ES 794

V = { (x,y,z,t) ∈ R⁴ | x=0, y=z-t }

W = span { (1,2,-1,0) }

V = ^(x=0) _(y=z-t)

W = s ⁽¹⁾ _{(2) (-1) (0)}

^(x=0) _{(y=z-t) (y=2x y=0) (z=-x z=0) (t=0)}

V ∩ W = (0,0,0,0)

ES 462

• ⁰₁ ⁰₁
• ¹₁ ⁰₀
• λ₂ = 1
• λ₁ + λ₂ = 0
• λ₁ = -1

ES 452

d (x, x²) = ||r - w|| = √< (r - w, r - w)> =

∫₀¹ (x - x²) (x - x²) dx = √∫₀¹ (x - x²)² dx = √∫₀¹ (x² + x⁴ - 2x³) dx =

√( [x³ / 3]₀¹ + [x⁵ / 5]₀¹ - 2 [x⁴ / 4]₀¹ ) = √( 1/3 + 1/5 - 1/2 ) = √( 10 + 6 - 15 / 30 ) = 1 / √30

ES 441

V = {f ∈ hom (R³, R³) | Im f ⊆ span (e₁)}

ES 608

f(x,y,z) = (x,0,x)

100 000 100 dim (V) = dim (Ker f) + dim (Im f)

ES 615

i00 0i0 00i

= -100 0-10 00-1

ES 633

P_i = (2,3,4)

P₀ = (1,0,0)

P₁ = (0,1,0)

P₂ = (0,0,2)

det -1-1x-1 10y 02z = 0

-1-1x-1 10y 02z = -1-1x-1 100 022 = 2(x-1) - ( -2y - z) =

= 2x-2 + 2y + z

2x + 2y + z - 2 = 0

d(P_i,π) = ^{|ax₀ + by₀ + cz₀ + d|} / _{√(a² + b² + c²)} = ^{|2∙2 + 2∙3 + 4 - 2|} / _√(4+4+1) = ¹² / ₃ = 4