Transformaciones Lineales
aldoayala03Práctica o problema6 de Marzo de 2023
7.304 Palabras (30 Páginas)141 Visitas
[pic 1]
``Año de la Unidad, Paz y el Desarrollo´´
[pic 2]
Curso:
Álgebra Lineal
Docente:
Sonia Casos Fernandez
Tema:
Transformaciones Lineales
Integrantes:
- Ayala Asto Aldo Esteban
- Alvarado Temoche Jhordan Michael
- Calle Huertas Oscar David
- Lopez Saavedra Angelina Yohana
- Ramos Chero Jefferson Alexander
- Determinar si las transformaciones dadas son lineales:
1) [pic 3]
[pic 4]
[pic 5]
[pic 6]
+[pic 7][pic 8]
[pic 9]
* No cumple con la cerradura respecto a la suma, por ende no es TI.
- [pic 10]
- [pic 11]
[pic 12]
( ++++....++++...+) [pic 13][pic 14][pic 15][pic 16][pic 17][pic 18][pic 19][pic 20]
* Si cumple[pic 21]
- [pic 22]
[pic 23]
[pic 24]
* Si cumple, es TL[pic 25]
- [pic 26]
No es TL[pic 27]
≠ [pic 28][pic 29]
Cuando se opera con , tenemos el espacio vectorial dado por x, para que se cumpla con las condiciones de TL, en este caso se debe partir de [pic 30]
- [pic 31]
- T[pic 32]
T[pic 33]
T[pic 34]
x)[pic 35]
+[pic 36][pic 37]
[pic 38]
+[pic 39][pic 40]
[pic 41]
[pic 42]
[pic 43]
[pic 44]
* Si es TI.[pic 45]
- [pic 46]
- [pic 47]
[pic 48]
[pic 49]
+[pic 50][pic 51]
+ [pic 52][pic 53]
- [pic 54]
[pic 55]
?
[pic 56]
[pic 57]
[pic 58]
*Si cumple que es L.I[pic 59]
- [pic 60]
- [pic 61]
[pic 62]
[pic 63]
[pic 64]
[pic 65]
[pic 66]
) = [pic 67][pic 68]
= [pic 69][pic 70]
[pic 71]
= [pic 72][pic 73]
* No cumple, T no es una TL.
- [pic 74]
))+(())[pic 75][pic 76]
([pic 77][pic 78]
[pic 79]
[pic 80]
[pic 81]
[pic 82]
[pic 83]
[pic 84]
[pic 85]
[pic 86]
[pic 87]
[pic 88]
- [pic 89][pic 90]
- Sean P(x), Q(x) [pic 91]
[pic 92]
[pic 93]
= [pic 94]
[pic 95]
[pic 96]
[pic 97]
=[pic 98]
- Sea [pic 99]
[pic 100]
[pic 101]
[pic 102]
[pic 103]
* De ambos T, es una TL
- [pic 104][pic 105]
- Sean f,g [pic 106]
[pic 107]
[pic 108]
[pic 109]
[pic 110]
[pic 111]
[pic 112]
- Sea [pic 113]
[pic 114]
[pic 115]
[pic 116]
[pic 117]
[pic 118]
[pic 119]
* De ambos T, es una TL.
- [pic 120][pic 121]
- [pic 122]
[pic 123]
[pic 124]
[pic 125]
[pic 126]
[pic 127]
= *Cumple[pic 128]
- [pic 129]
[pic 130]
= [pic 131]
= [pic 132]
* Para los guiones, Tes una TL
- [pic 133]
. [pic 134]
[pic 135]
[pic 136]
[pic 137]
[pic 138]
[pic 139]
.[pic 140]
[pic 141]
[pic 142]
[pic 143]
[pic 144]
[pic 145]
T es una transformación lineal.
12. [pic 146]
[pic 147]
[pic 148]
[pic 149]
T no es una transformación lineal.
- Hallar 𝑁(𝑇), 𝐼𝑚(𝑇), 𝜂(𝑇), 𝜌(𝑇) 𝑦 𝐴𝑇
- 𝑇: ℝ4→ℝ2, 𝑇 (𝑥, 𝑦, 𝑧, 𝑤) = (𝑥+𝑧, 𝑦+𝑤)
(𝑥+𝑧, 𝑦+𝑤) = (a, b)
[pic 150][pic 151]
[pic 152][pic 153]
Por lo tanto: I𝑚 (T) = ℝ2
(𝑥+𝑧, 𝑦+𝑤) = (0, 0)
[pic 154][pic 155]
[pic 156]
[pic 157]
[pic 158]
Por lo tanto: 𝑁(𝑇) = {(x, y, z, w) ℝ4 / x=z}[pic 159]
= gen {(0,0)} dim(N(T)) = 1[pic 160]
- 𝑇: ℝ2→ℝ3, 𝑇 (𝑥, 𝑦) = (𝑥,2𝑦, 𝑥−𝑦)
(𝑥, 2y, x-y) = (a, b, c)
[pic 161]
Por lo tanto: I𝑚 (T) = ℝ3
(𝑥, 2y, x-y) = (0, 0, 0)
[pic 162]
[pic 163]
[pic 164]
[pic 165]
Por lo tanto: 𝑁(𝑇) = {(x, y) ℝ2 / x = 0, 2y = 0 -y = 0}[pic 166][pic 167]
- 𝑇: ℝ2→ℝ2, 𝑇 (𝑥, 𝑦) = (𝑥+𝑦, 𝑥−𝑦) 𝐵1={(1,2),(−2,3)} 𝐵2={(1,1),(−1,0)}
(𝑥+y, x-y) = (a, b)
[pic 168][pic 169]
Por lo tanto: I𝑚 (T) = ℝ2
(𝑥+y, x-y) = (0, 0)
[pic 170][pic 171]
[pic 172]
[pic 173]
Por lo tanto: 𝑁(𝑇) = {(x, y) ℝ2 / x + y = 0 -2y = 0}[pic 174][pic 175]
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