Esercizi geometria

LEZ. 4: sistemi

x - 2y + 3z + t = 22x - 5y + 3z = 5x - y + 3t = 1

A =

rang(A) =

rang(A, B) = 2

M =

x =

y =

S = {(3h - 5k, 3h - 2k - 1, h, k) | h, k ∈ &Reals;}

LEZ. 4: sistemi

x-2y+3z+t=22x-5y+3z=5x-y+4y+3t=1

A = ^{1 -2 3|2}        _{2 -5 3|0}               ^{1 -4 0|3}

A,B = ^{1 -2 3 0 2}             _{2 -5 3 0 5}                  ^{1 -4 0 3 1}

rang (A) = ^{1 -2 3}             _{-2 -5 3}             ^{1 -4 0} = (-18-6)-(-15-9)=-24+24=0

= ^{-2 3 0}         _{-5 3 0}         ^{-4 0 3} = (-15-2)-(-5-12)=0

= ^{2 -3 0}         _{3 0 3}        ^{2 3 3} = (-54, -9-6x5)=0

= ^{2 -2 0}       _{2 0 3}   ^{3 0 3} = (27)-(3+18)=0

raug (A) = 1, raug (A,B) = 2 il sistema è compatibile

det = -5 - (-u) = 5t(u=-1)

x-2y=2 -3h-k

2x-5y=-9h+5

x= ^{(-5)(2-3h-k)-(2)(5-9h)}                                                _-1

= 5k-3h -1 = 3h-5k

y= ^{(5-9h)-(2)(2-3h-k)}                                                                                                                                                                                                                                                                        

^3h-2k-1

sistema omogeneo -> sempre compatibile

A = ( 1   3   3   2 ) ( 2   6   9   5 ) ( 0  -3   3   5 )

M = ( 3   3 ) ( 6   9 ) = 27 - 18 = 9

H₀ = ( 1   3   3 ) ( 2   6   9 ) ( 0  -3   3 ) = (-18-18)-(-27+18) = 9

rang(A) = rang(A,B) = 3

{ x + 3y + 3z = -2k 2x + 6y + 9z = -5k -3y + 3z - t = k

X = (  -2k   3   3     ) (  -5k   6   9   ) ( k   -3   3     ) =   (18(-2k) + ... + 9(-8k))   (9     )

z = -36k + 27k + u5k -18k -8-5k +u5k = 9k/9 = k

{ y = ( 1  -2k   3 ) ( 2  -5k   9 ) ( 0   k   3 ) = -15k + 6k - (9k - 12k) = -6/9 k = -2/3 k

{ a = ( 4   3  -2k   ) ( 0  -3   -5k )

= ( 6k +12k - (15k + 6k) ) / 9 =   3k/9 = -1/3 k

S = { (k, -2/3 k, -1/3 k, k) | k ∈ ℝ }

Scansionato con CamScanner

Lez. 8: Sottospazio Vettoriale, Base e Dimensione

S={(x,y,z)∈ℝ³|2x-y+z=0}

S={(x,2x+z,z)|x,z∈ℝ³}

Sottospazio vettoriale.

0∈S: per x=0