03 em 05 empe u3 10

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V 𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 V = ℒ(𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 ) . P3 [𝑥]

𝑥

P3 [𝑥] = ℒ(p1 (𝑥), p2 (𝑥), p3 (𝑥), p4 (𝑥)) , p1 (𝑥) = 1, p2 (𝑥) = 𝑥, p3 (𝑥) = 𝑥 2 ,

P[𝑥]

p4 (𝑥) = 𝑥 3 .

𝑥

V 𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 V 𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 ℝ𝑛

ℒ(𝑒⃗1 , 𝑒⃗2 , … , 𝑒⃗𝑛 ) 𝑒⃗1 = (1,0,0, … ,0,0) 𝑒⃗2 = (0,1,0, … ,0,0) ⋮ 𝑒⃗𝑛 = (0,0,0, … ,0,1)


ℝ𝑛

ℝ𝑛

𝑥⃗ = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝑥1 𝑒⃗1 + 𝑥2 𝑒⃗1 + ⋯ + 𝑥𝑛 𝑒⃗𝑛 .

𝛼1 , 𝛼2 , … , 𝛼𝑛 ⃗​⃗ , 𝛼1 𝑒⃗1 + 𝛼2 𝑒⃗1 + ⋯ + 𝛼𝑛 𝑒⃗𝑛 = 0

𝛼1 𝑒⃗1 + 𝛼2 𝑒⃗1 + ⋯ + 𝛼𝑛 𝑒⃗𝑛 = (𝛼1 , 𝛼2 , … , 𝛼𝑛 ) = ⃗0⃗ , 𝛼1 = 𝛼2 = ⋯ = 𝛼𝑛 = 0

P3 [𝑥] V V

{𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , … , 𝑢 ⃗​⃗𝑛 }

V {𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑚 }

V

𝑛=𝑚 V V

V 𝐝𝐢𝐦 𝐕 V V 𝑘

𝑘 V

dim V = 𝑤 ⃗​⃗​⃗1 , 𝑤 ⃗​⃗​⃗2 , … , 𝑤 ⃗​⃗​⃗𝑘


V

𝑘 𝑘 V

V 𝑘+1

𝑘

V

𝑢 ⃗​⃗1 = (𝑎11 , 𝑎21 , … , 𝑎𝑚1 ) 𝑢 ⃗​⃗2 = (𝑎12 , 𝑎22 , … , 𝑎𝑚2 ) ⋮ 𝑢 ⃗​⃗𝑛 = (𝑎1𝑛 , 𝑎2𝑛 , … , 𝑎𝑚𝑛 ) ℝm

𝑛

dim ℝm = 𝑚

𝑛>𝑚

𝑛>𝑚

𝑥1 , 𝑥2 , … , 𝑥𝑛

⃗​⃗ , 𝑥1 𝑢 ⃗​⃗1 + 𝑥2 𝑢 ⃗​⃗2 + ⋯ + 𝑥𝑛 𝑢 ⃗​⃗𝑛 = 0

𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 0 𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 0 ⋮ 𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 = 0 𝑛>𝑚

V

𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 V = ℒ(𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 ).

V


𝑣⃗

V ⃗​⃗ = 𝑥1 𝑣⃗1 + 𝑥2 𝑣⃗2 + ⋯ + 𝑥𝑛 𝑣⃗𝑛 , v 𝑥1 , 𝑥2 , … , 𝑥𝑛 (𝑥1 , 𝑥2 , … , 𝑥𝑛 )

{𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 }

⃗​⃗ v ⃗​⃗ v

𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 ℒ(𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 ) ℒ(𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 )

𝑥1 𝑣⃗1 + 𝑥2 𝑣⃗2 + ⋯ + 𝑥𝑛 𝑣⃗𝑛 {𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 }

ℝ4 𝑢 ⃗​⃗1 = (2, −1,3,1) 𝑢 ⃗​⃗2 = (1,1,4,2) 𝑢 ⃗​⃗3 = (3, −3,2,0) ℒ(𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 )

𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3

ℒ(𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 )

𝑣⃗

𝑣⃗ = 𝑥𝑢 ⃗​⃗1 + 𝑦𝑢 ⃗​⃗2 + 𝑧𝑢 ⃗​⃗3 . 𝑥, 𝑦,

𝑧

𝑣⃗

𝑥 = 1, 𝑦 = 2

𝑧 = −1

𝑣⃗ = 𝑥𝑢 ⃗​⃗1 + 𝑦𝑢 ⃗​⃗2 + 𝑧𝑢 ⃗​⃗3 = (2, −1,3,1) + 2(1,1,4,2) − (3, −3,2,0) = (1,4,9,5). 𝑥, 𝑦, 𝑥 = 3, 𝑦 = 1

𝑧 = −2

𝑧


𝑣⃗ = 𝑥𝑢 ⃗​⃗1 + 𝑦𝑢 ⃗​⃗2 + 𝑧𝑢 ⃗​⃗3 = 3(2, −1,3,1) + (1,1,4,2) − 2(3, −3,2,0) = (1,4,9,5). 𝑣⃗

𝑣⃗ = (5 − 2𝜆)𝑢 ⃗​⃗1 + 𝜆𝑢 ⃗​⃗2 + (𝜆 − 3)𝑢 ⃗​⃗3 , 𝜆

ℝ4 𝑢 ⃗​⃗1 = (2,4,3,4) 𝑢 ⃗​⃗2 = (2,0,1,4) 𝑢 ⃗​⃗3 = (1, −1,2,6)

𝑣⃗

ℒ(𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 ) ℒ(𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 )

𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3

𝑣⃗

𝑣⃗ = 𝑥1 𝑢 ⃗​⃗1 + 𝑥2 𝑢 ⃗​⃗2 + 𝑥3 𝑢 ⃗​⃗3 = 𝑦1 𝑢 ⃗​⃗1 + 𝑦2 𝑢 ⃗​⃗2 + 𝑦3 𝑢 ⃗​⃗3 ,

𝑣⃗ (𝑥1 − 𝑦1 )𝑢 ⃗​⃗1 + (𝑥2 − 𝑦2 )𝑢 ⃗​⃗2 + (𝑥3 − 𝑦3 )𝑢 ⃗​⃗3 = 0 , 𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 𝑥1 − 𝑦1 = 0, 𝑥1 = 𝑦1 = 𝑥2 = 𝑦2 = 𝑥3 = 0

𝑥2 − 𝑦2 = 0 , 𝑣⃗

𝑥3 − 𝑦3 = 0 ,


ℒ(𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 ) 𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛

𝑣⃗ 𝑥2 𝑣⃗2 + ⋯ + 𝑥𝑛 𝑣⃗𝑛

V V

𝑥1 𝑣⃗1 +

𝑛

𝒰 = {𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , … , 𝑢 ⃗​⃗𝑛 }

𝒱 = {𝑣⃗1 , 𝑣⃗2 , … , 𝑣⃗𝑛 }

𝑣⃗1 = 𝑎11 𝑢 ⃗​⃗1 + 𝑎21 𝑢 ⃗​⃗2 + ⋯ + 𝑎𝑛1 𝑢 ⃗​⃗𝑛 𝑣⃗2 = 𝑎12 𝑢 ⃗​⃗1 + 𝑎22 𝑢 ⃗​⃗2 + ⋯ + 𝑎𝑛2 𝑢 ⃗​⃗𝑛 ⋮ 𝑣⃗𝑛 = 𝑎1𝑛 𝑢 ⃗​⃗1 + 𝑎2𝑛 𝑢 ⃗​⃗2 + ⋯ + 𝑎𝑛𝑛 𝑢 ⃗​⃗𝑛

𝑎11 𝑎12 ⋯ 𝑎1𝑛 𝑎11 𝑎21 ⋯ 𝑎𝑛1 T 𝑎 𝑎22 ⋯ 𝑎2𝑛 𝑎 𝑎22 ⋯ 𝑎𝑛2 A = ( 21 ) = ( 12 ) ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 𝑎𝑛1 𝑎𝑛2 ⋯ 𝑎𝑛𝑛 𝑎1𝑛 𝑎2𝑛 ⋯ 𝑎𝑛𝑛 𝒰

𝒱

𝑣⃗1 𝑢 ⃗​⃗1 𝑣⃗2 𝑢 ⃗​⃗2 ( ) = A( ) ⋮ ⋮ 𝑣⃗𝑛 𝑢 ⃗​⃗𝑛 𝑣⃗

V 𝑣⃗ = 𝛼1 𝑢 ⃗​⃗1 + 𝛼2 𝑢 ⃗​⃗2 + ⋯ + 𝛼𝑛 𝑢 ⃗​⃗𝑛 = 𝛽1 𝑣⃗1 + 𝛽2 𝑣⃗2 + ⋯ + 𝛽𝑛 𝑣⃗𝑛 ,

𝒰 = {𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 } {𝑣⃗1 , 𝑣⃗2 , 𝑣⃗3 }

(𝛼1 , 𝛼2 , … , 𝛼𝑛 ) (𝛽1 , 𝛽2 , … , 𝛽𝑛 )

𝑣⃗ = 𝛽1 𝑣⃗1 + 𝛽2 𝑣⃗2 + ⋯ + 𝛽𝑛 𝑣⃗𝑛

𝒱=


𝑣⃗𝑖

𝑢 ⃗​⃗𝑖

𝑣⃗ = 𝛽1 (𝑎11 𝑢 ⃗​⃗1 + 𝑎21 𝑢 ⃗​⃗2 + ⋯ + 𝑎𝑛1 𝑢 ⃗​⃗𝑛 ) + 𝛽2 (𝑎12 𝑢 ⃗​⃗1 + 𝑎22 𝑢 ⃗​⃗2 + ⋯ + 𝑎𝑛2 𝑢 ⃗​⃗𝑛 ) + ⋯ + 𝛽𝑛 (𝑎1𝑛 𝑢 ⃗​⃗1 + 𝑎2𝑛 𝑢 ⃗​⃗2 + ⋯ + 𝑎𝑛𝑛 𝑢 ⃗​⃗𝑛 ) = (𝑎11 𝛽1 + 𝑎12 𝛽2 + ⋯ + 𝑎1𝑛 𝛽𝑛 )𝑢 ⃗​⃗1 + (𝑎21 𝛽1 + 𝑎22 𝛽2 + ⋯ + 𝑎𝑛𝑛 𝛽𝑛 )𝑢 ⃗​⃗2 + ⋯ + (𝑎𝑛1 𝛽1 + 𝑎𝑛2 𝛽2 + ⋯ + 𝑎𝑛𝑛 𝛽𝑛 )𝑢 ⃗​⃗𝑛 𝑣⃗

𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , … , 𝑢 ⃗​⃗𝑛 𝑣⃗ = 𝛼1 𝑢 ⃗​⃗1 + 𝛼2 𝑢 ⃗​⃗2 + ⋯ + 𝛼𝑛 𝑢 ⃗​⃗𝑛

𝛼1 = 𝑎11 𝛽1 + 𝑎12 𝛽2 + ⋯ + 𝑎1𝑛 𝛽𝑛 𝛼2 = 𝑎21 𝛽1 + 𝑎22 𝛽2 + ⋯ + 𝑎𝑛𝑛 𝛽𝑛 ⋮

𝛼𝑛 = 𝑎𝑛1 𝛽1 + 𝑎𝑛2 𝛽2 + ⋯ + 𝑎𝑛𝑛 𝛽𝑛

𝛽1 𝛼1 𝑎11 𝑎12 ⋯ 𝑎1𝑛 𝛼2 𝑎21 𝑎22 ⋯ 𝑎2𝑛 𝛽2 ( )=( )( ) ⋮ ⋮ ⋮ ⋮ ⋮ 𝛼𝑛 𝑎𝑛1 𝑎𝑛2 ⋯ 𝑎𝑛𝑛 𝛽𝑛 𝛽1 𝛼1 𝛼2 𝛽2 ( ) = A( ) ⋮ ⋮ 𝛼𝑛 𝛽𝑛 A

𝒰

𝒱

𝒰 𝒱

𝒱 𝒰

−1

A

I

𝛽1 𝛽1 𝛼1 𝛼2 𝛽2 𝛽2 −1 ( ) = A A( ) = I( ) ⋮ ⋮ ⋮ 𝛼𝑛 𝛽𝑛 𝛽𝑛

A


𝛽1 𝛼1 𝛽2 −1 𝛼2 ( )=A ( ). ⋮ ⋮ 𝛼𝑛 𝛽𝑛 A−1 −1

A

ℝ3 𝒰 = {𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 } 𝑢 ⃗​⃗1 = (4, −1,2) 𝑢 ⃗​⃗2 = (2,3,5) 𝑢 ⃗​⃗3 = (1, −1,1) 𝒱 = {𝑣⃗1 , 𝑣⃗2 , 𝑣⃗3 }

𝑣⃗1 = (9, −9,5) 𝑣⃗2 = (5,3,6) 𝑣⃗3 = (8, −7,1) ℝ3

𝒰 = {𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 } 𝑢 ⃗​⃗𝑗 ℝ3

𝑣⃗𝑖 𝒱 = {𝑣⃗1 , 𝑣⃗2 , 𝑣⃗3 }

𝑢 ⃗​⃗𝑗

𝑣⃗𝑖

𝑣⃗1 = 3𝑢 ⃗​⃗1 − 2𝑢 ⃗​⃗2 + 𝑢 ⃗​⃗3 , 𝑣⃗2 = 𝑢 ⃗​⃗1 + 𝑢 ⃗​⃗2 − 𝑢 ⃗​⃗3 , 𝑣⃗3 = 2𝑢 ⃗​⃗1 − 𝑢 ⃗​⃗2 + 2𝑢 ⃗​⃗3 .

3 −2 1 Ρ = (1 1 −1) 2 −1 2

{𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 }

𝒱 = {𝑣⃗1 , 𝑣⃗2 , 𝑣⃗3 } P

P

3 1 2 P𝑇 = A = (−2 1 −1) 1 −1 2

𝒰 = {𝑢 ⃗​⃗1 , 𝑢 ⃗​⃗2 , 𝑢 ⃗​⃗3 } 𝒰 𝒱

𝒰= 𝒱


đ?‘Łâƒ— (đ?›ź1 , đ?›ź2 , đ?›ź3 ) (đ?›˝1 , đ?›˝2 , đ?›˝3 )

â„?3 đ?’° = {đ?‘˘ ⃗⃗1 , đ?‘˘ ⃗⃗2 , đ?‘˘ ⃗⃗3 } đ?’ą = {đ?‘Łâƒ—1 , đ?‘Łâƒ—2 , đ?‘Łâƒ—3 }

đ?‘Łâƒ— = đ?›˝1 đ?‘Łâƒ—1 + đ?›˝2 đ?‘Łâƒ—2 + đ?›˝3 đ?‘Łâƒ—3 đ?‘Łâƒ—đ?‘–

đ?‘˘ ⃗⃗đ?‘–

đ?‘Łâƒ— = đ?›˝1 (3đ?‘˘ ⃗⃗1 − 2đ?‘˘ ⃗⃗2 + đ?‘˘ ⃗⃗3 ) + đ?›˝2 ( đ?‘˘ ⃗⃗1 + đ?‘˘ ⃗⃗2 − đ?‘˘ ⃗⃗3 ) + đ?›˝3 (2đ?‘˘ ⃗⃗1 − đ?‘˘ ⃗⃗2 + 2đ?‘˘ ⃗⃗3 ) (3đ?›˝ )đ?‘˘ (−2đ?›˝ )đ?‘˘ = ⃗⃗3 1 + đ?›˝2 + 2đ?›˝3 ⃗⃗1 + 1 + đ?›˝2 − đ?›˝3 ⃗⃗2 + (đ?›˝1 − đ?›˝2 + 2đ?›˝3 )đ?‘˘ đ?‘Łâƒ—

đ?‘˘ ⃗⃗1 , đ?‘˘ ⃗⃗2 , đ?‘˘ ⃗⃗3 đ?‘Łâƒ— = đ?›ź1 đ?‘˘ ⃗⃗1 + đ?›ź2 đ?‘˘ ⃗⃗2 + đ?›ź3 đ?‘˘ ⃗⃗3

đ?›ź1 = 3đ?›˝1 + đ?›˝2 + 2đ?›˝3 đ?›ź2 = −2đ?›˝1 + đ?›˝2 − đ?›˝3 đ?›ź3 = đ?›˝1 − đ?›˝2 + 2đ?›˝3

đ?›ź1 3 1 2 đ?›˝1 (đ?›ź2 ) = (−2 1 −1) (đ?›˝2 ) đ?›ź3 1 −1 2 đ?›˝3 đ?›˝1 = A (đ?›˝2 ) đ?›˝3 đ?’°

đ?’ą

đ?’° đ?’ą

đ?’ą đ?’°

A

đ?›ź1 đ?›˝1 −1 đ?›ź (đ?›˝2 ) = A ( 2 ) . đ?›ź3 đ?›˝3 Muy bien, has terminado la revisiĂłn del material de estudio de las tres unidades de aprendizaje de este mĂłdulo, ÂżTienes alguna duda? Contacta al docente o plantea tus dudas colectivas en el foro destinado para dicho fin.


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