U1_Espacios_vectoriales

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ÍNDICE 
 
 
 
 
 
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ℝ2
ℝ3
 
 
 
 
https://www.youtube.com/watch?v=A46dFgcyRvQ
https://www.youtube.com/watch?v=A46dFgcyRvQ
 
 
Cambios en la 
temperatura de 
la Tierra
Termodinámica
Espacios 
vectoriales
Modelación 
matemática
Funciones 
de varias 
variables
Derivadas 
parciales
 
 
https://www.youtube.com/watch?v=sDo7saKaEys
https://www.youtube.com/watch?v=sDo7saKaEys
 
𝑨 |𝑨| ‖𝑨‖
‖ ‖ | |
https://www.youtube.com/watch?v=guk7RQY2rIY
https://www.youtube.com/watch?v=guk7RQY2rIY
 
 
Figura 1. Vector y matrices 
1 −1
𝑨 + (−𝑩)
Figura 2. Vectores, determinantes y planos
http://www.educaplus.org/play-289-Suma-gr%C3%A1fica-de-vectores.html
 
𝑂
𝑂 = (0,0) 𝑂 = (0,0,0) 𝑋𝑌 𝑨
(𝑎1, 𝑎2)
𝑨 𝑨
𝑨 = 〈𝑎1, 𝑎2〉
(𝑎1, 𝑎2)
https://www.youtube.com/watch?v=qvw7j9eKGdg
https://www.youtube.com/watch?v=qvw7j9eKGdg
 
Figura 3. Suma de vectores
𝒊 𝒋 𝒊 = 〈1,0〉 𝒋 = 〈0,1〉
�̂� 𝒋̂
𝒆𝟏 y 𝒆𝟐 o 𝑒1̅ y 𝑒2̅ 𝒊 =
〈1,0,0〉 𝒋 = 〈0,1,0〉 𝒌 = 〈0,0,1〉 (𝑎1, 𝑎2)
〈𝑎1, 𝑎2〉 = 𝑎1𝒊 + 𝑎2𝒋 𝑨 = 𝑎1𝒊 + 𝑎2𝒋 𝑎1 𝑎2
𝒊 𝒋 𝑨 �⃗⃗� = 𝑶𝑷⃗⃗⃗⃗⃗⃗ 
𝑃 𝑷 𝑷 = �⃗⃗� .
𝑨 = 𝑎1𝒊 + 𝑎2𝒋 𝑩 = 𝑏1𝒊 + 𝑏2𝒋
|𝑨| = √𝑎12 + 𝑎22
𝑨 + 𝑩 = (𝑎1 + 𝑏1)𝒊 + (𝑎2 + 𝑏2)𝒋 〈𝑎1, 𝑎2〉 + 〈𝑏1, 𝑏2〉 = 〈𝑎1 + 𝑏1, 𝑎2 + 𝑏2〉
𝑨 − 𝑩 = (𝑎1 − 𝑏1)𝒊 + (𝑎2 − 𝑏2)𝒋 〈𝑎1, 𝑎2〉 − 〈𝑏1, 𝑏2〉 = 〈𝑎1 − 𝑏1, 𝑎2 − 𝑏2〉
𝜆𝑨 = 𝝀(𝑎1𝒊 + 𝑎2𝒋) = 𝝀𝑎1𝒊 + 𝝀𝑎2𝒋 𝜆〈𝑎1, 𝑎2〉 = 〈𝜆𝑎1, 𝜆𝑎2〉
 
 
Figura 4. Vista Geométrica
 
𝑨 = 〈𝑎1, 𝑎2, 𝑎3〉
 
Figura 5. Representación del vector
 
〈𝑎1, 𝑎2, 𝑎3〉 = 𝑎1𝒊 + 𝑎2𝒋 + 𝑎3𝒌
 
𝑨 = 〈𝑎1, 𝑎2, 𝑎3〉 𝑩 = 〈𝑏1, 𝑏2, 𝑏3〉 〈𝑎1, 𝑎2, 𝑎3〉 + 〈𝑏1, 𝑏2, 𝑏3〉 =
〈𝑎1 + 𝑏1, 𝑎2 + 𝑏2, 𝑎3 + 𝑏3〉
‖𝑎1𝒊 + 𝑎2𝒋 + 𝑎3𝒌‖ = ‖〈𝑎1, 𝑎2, 𝑎3〉‖ = √𝑎1
2 + 𝑎2
2 + 𝑎3
2
𝑟 = √𝑎12 + 𝑎22 |𝑨| =
√𝑟2 + 𝑎32 √𝑎12 + 𝑎22 + 𝑎32
 
Figura 6. Representación de vectores 
https://www.youtube.com/watch?v=qLD3wWcF6eA
https://www.youtube.com/watch?v=qLD3wWcF6eA
 
�̂�
〈3,4〉
1
5
〈3,4〉 = 〈
3
5
+
4
5
〉
‖〈3,4〉 ‖ = 5
𝑨/|𝑨|
ℝ𝑛 𝑛
𝑛
 𝑨
 �⃗⃗� 
 �̂�
 �̅� 𝑣 
 
𝑨𝟏 𝑷𝟏⃗⃗⃗⃗ ⃗ 𝑢1̅̅ ̅ 𝑣1⃗⃗⃗⃗ 
 
 
 
Figura 7. Fuerza resultante de dos fuerzas en distintas direcciones 
 
𝑓(𝑥) = 4𝑥3 + 𝑥2 − 8𝑥
𝑔(𝑥) = 4𝑥3 ℎ(𝑥) = 𝑥2 𝑖(𝑥) = −8𝑥 𝑓(𝑥) = 𝑔(𝑥) + ℎ(𝑥) +
𝑖(𝑥)
 
ℝ2 ℝ3
http://www.youtube.com/watch?v=wQfIndxiGDI
 
 + �̅� �̅� �̅� +
�̅�
 𝜆
�̅� 𝜆�̅�
 𝑉
�̅� �̅� 𝑉 �̅� + �̅� ∈ 𝑉
 𝑉
𝑢1̅̅ ̅ 𝑢2̅̅ ̅ 𝑢3̅̅ ̅ ∈ 𝑉 𝑢1̅̅ ̅ + 𝑢2̅̅ ̅) + 𝑢3̅̅ ̅ = 𝑢1̅̅ ̅ + (𝑢2̅̅ ̅ + 𝑢3̅̅ ̅)
 �̅� ∈ 𝑉 ∃ 0̅ ∈ 𝑉
0̅ + �̅� = �̅� + 0̅ = �̅�
 𝑉
�̅� ∈ 𝑉 −�̅� ∈ 𝑉 −�̅� + �̅� = �̅� + (−�̅�) = 0̅
 𝑢1̅̅ ̅ 𝑢2̅̅ ̅ ∈ 𝑉 𝑢1̅̅ ̅ +
𝑢2̅̅ ̅ = 𝑢2̅̅ ̅ + 𝑢1̅̅ ̅
 𝑉 𝜆 ∈
 
ℝ �̅� ∈ 𝑉 𝜆�̅� ∈ 𝑉
 
𝜆 ∈ ℝ 𝑢1̅̅ ̅ 𝑢2̅̅ ̅ ∈ 𝑉 𝜆(𝑢1̅̅ ̅ + 𝑢2̅̅ ̅) = 𝜆𝑢1̅̅ ̅ + 𝜆𝑢2̅̅ ̅
 
𝜆, 𝜇 ∈ ℝ �̅� ∈ 𝑉
(𝜆 + 𝜇)�̅� = 𝜆�̅� + 𝜇�̅�
 𝜆, 𝜇 ∈ ℝ
�̅� ∈ 𝑉 𝜆𝜇(�̅�) = 𝜆(𝜇�̅�)
 �̅� ∈ 𝑉
1�̅� = �̅�
ℝ2 ℝ3
https://www.youtube.com/watch?v=q6IQJA8qvok
https://www.youtube.com/watch?v=q6IQJA8qvok
 
𝑋𝑌 𝑌𝑍 𝑍𝑋
𝑈 𝑉 𝑉 𝑈
𝑉
𝑉
𝑈 𝑉
𝑉
 𝑈
 𝜆, 𝜇 ∈ ℝ �̅� �̅� 𝑉 𝜆�̅� + 𝜇�̅� ∈ 𝑈
https://www.youtube.com/watch?v=z4-Ki7Bdh4w
https://www.youtube.com/watch?v=z4-Ki7Bdh4w
 
𝑈
𝑈 = {𝜆�̅�|�̅� ≠ 0 ̅, 𝜆 ∈ ℝ}
𝑉
 0̅ 𝜆 = 0 𝜆�̅� = 0̅
�̅� ∈ 𝑈 ⊂ 𝑉
 〈𝑎, 𝑏, 𝑐〉 𝑈
𝑈 𝛼(𝜆〈𝑎, 𝑏, 𝑐〉) + 𝛽(𝜇〈𝑎, 𝑏, 𝑐〉) = 𝛼𝜆〈𝑎, 𝑏, 𝑐〉 + 𝛽𝜇〈𝑎, 𝑏, 𝑐〉 =
(𝛼𝜆 + 𝛽𝜇)〈𝑎, 𝑏, 𝑐〉 ∈ 𝑈
𝑈
ℒ〈𝑎,𝑏,𝑐〉
𝜆�̅� + 𝜇�̅� �̅� �̅�
𝑉
𝜆1𝑣1̅̅ ̅ + 𝜆2𝑣2̅̅ ̅ + ⋯+ 𝜆𝑛𝑣𝑛̅̅ ̅,
𝜆𝑖 ∈ ℝ 𝑣�̅� ∈ 𝑉 𝑖 = 1, 2, 3, … , 𝑛
 
�̅� �̅� ∈ ℝ3 𝑈 = {𝜆�̅� + 𝜇�̅�|�̅�, �̅� ≠ 0 ̅, �̅� ≠ 𝛼�̅� ∀𝛼 ∈ ℝ y 𝜆, 𝜇 ∈ ℝ}
ℝ3
 𝑈 ≠ ∅ ℝ3 𝜆 = 1 𝜇 =
0 �̅� ∈ 𝑈 𝜆 = 0 𝜇 = 1 �̅� ∈ 𝑈
 �̅� �̅� 𝛼(𝜆1�̅� + 𝜇1�̅�) +
𝛽(𝜆2�̅� + 𝜇2�̅�) (𝛼𝜆1�̅� + 𝛼𝜇1�̅�) +
(𝛽𝜆2�̅� + 𝛽𝜇2�̅�) = (𝛼𝜆1 + 𝛽𝜆2)�̅� + (𝛼𝜇1 + 𝛽𝜇2)�̅�
𝑈
𝑈
𝒫𝑢,�̅�
https://www.youtube.com/watch?v=pGwhr3j0Zx4
https://www.youtube.com/watch?v=pGwhr3j0Zx4
 
𝒜 = {𝜆1𝑣1̅̅ ̅ + 𝜆2𝑣2̅̅ ̅ + ⋯+ 𝜆𝑘𝑣𝑘̅̅ ̅|𝑣�̅� ≠ 0 ̅, 𝑣�̅� ≠ 𝛼𝑣�̅� ∀𝛼 ∈ ℝ y ∀ 𝑖, 𝑗 = 1, 2, 3, … , 𝑘, con 𝜆𝑖 ∈ ℝ, 𝑘
< 𝑛}
ℝ𝑛 𝑣1̅̅ ̅, 𝑣2̅̅ ̅, … , 𝑣𝑘̅̅ ̅
𝑈 = {𝑣1̅̅ ̅, 𝑣2̅̅ ̅, … , 𝑣𝑚̅̅ ̅̅ } 𝑉
𝜆1𝑣1̅̅ ̅ + 𝜆2𝑣2̅̅ ̅ + ⋯+ 𝜆𝑚𝑣𝑚̅̅ ̅̅ = 0 ̅
𝜆1 = 𝜆2 = ⋯ = 𝜆𝑚 = 0
 𝑈 𝜆1𝑣1̅̅ ̅ + 𝜆2𝑣2̅̅ ̅ + ⋯+ 𝜆𝑚𝑣𝑚̅̅ ̅̅ = 0 ̅
0 𝜆𝑖 = 0 𝑖
 
 
ℝ3
{𝒊, 𝒋, 𝒌}
https://www.youtube.com/watch?v=hEwMcCd-57o
https://www.youtube.com/watch?v=hEwMcCd-57o
https://www.youtube.com/watch?v=QWFk9nT5fBk
https://www.youtube.com/watch?v=hEwMcCd-57o
 
ℝ
ℝ2
ℝ3
�̅� = 〈𝑢1, 𝑢2, 𝑢3〉 �̅� =
〈𝑣1, 𝑣2, 𝑣3〉
�̅� ∙ �̅� = 𝑢1𝑣1 + 𝑢2𝑣2 + 𝑢3𝑣3
�̅� = �̅� − �̅�
 
 
Figura 8. Producto punto o 
escalar 
‖�̅�‖2 = ‖�̅�‖2 + ‖�̅�‖2 − 2‖�̅�‖‖�̅�‖ cos 𝜃 2‖�̅�‖‖�̅�‖ cos 𝜃 =
‖�̅�‖2 + ‖�̅� ‖2 − ‖�̅�‖ 2 ‖�̅�‖2 = √𝑢12 + 𝑢22 + 𝑢22
2
‖�̅� ‖2 ‖�̅�‖ 2 �̅�
〈𝑢1 − 𝑣1, 𝑢2 − 𝑣2, 𝑢3 − 𝑣3〉
‖�̅�‖2 + ‖�̅�‖2 − ‖�̅�‖2 = 2(𝑢1𝑣1 + 𝑢2𝑣2 + 𝑢3𝑣3)
‖�̅�‖‖�̅�‖ cos 𝜃 = 𝑢1𝑣1 + 𝑢2𝑣2 + 𝑢3𝑣3 = �̅� ∙ �̅�
 
�̅� ∙ �̅� = ‖�̅�‖‖�̅�‖ 𝐜𝐨𝐬 𝜽
https://www.youtube.com/watch?v=8OEllQgiyN
https://www.youtube.com/watch?v=8OEllQgiyN
https://www.youtube.com/watch?v=bAxlqrEhHeY
https://www.youtube.com/watch?v=bAxlqrEhHeY
https://www.youtube.com/watch?v=8OEllQgiyNI
 
𝜃 = 𝑐𝑜𝑠−1 (
�̅� ∙ �̅�
‖�̅�‖‖�̅�‖
)
𝜋/2
cos
𝜋
2
= 0
https://www.youtube.com/watch?v=gB6Q7rxM3d0
https://www.youtube.com/watch?v=2PJDDsLqRDc
https://www.youtube.com/watch?v=gB6Q7rxM3d0
https://www.youtube.com/watch?v=gB6Q7rxM3d0
 
https://www.youtube.com/watch?v=FjwSKiQ2uIM
https://www.youtube.com/watch?v=FjwSKiQ2uIM
https://www.youtube.com/watch?v=r_J5z6gAMzE
https://www.youtube.com/watch?v=r_J5z6gAMzE
https://www.youtube.com/watch?v=Gq3fyrZt2yw
https://www.youtube.com/watch?v=Gq3fyrZt2yw
https://www.youtube.com/watch?v=FQwE5x1CkNw
https://www.youtube.com/watch?v=FQwE5x1CkNw
 
𝑋
𝑌 𝒊 o 𝒋 1
𝒗 𝑭
 
Figura 9. Fuerza F 
𝑝𝑟𝑜𝑗𝑭𝒗 = ‖𝑭‖ 𝑐𝑜𝑠 𝜃 =
𝑭 ∙ 𝒗
‖𝒗‖
http://www.youtube.com/watch?v=S2Ri8FaP7zo&index=8&list=PLVEkI8DcwbMtXW1Ug8HklcTKLzoiDVn2F
https://www.youtube.com/watch?v=uqT0Wc4lgt4
 
ℝ2 
ℝ3.
ℝ3.
2 × 2 
|
𝑎 𝑏
𝑐 𝑑
| = 𝑎𝑑 − 𝑏𝑐
3 × 3
|
𝑎1 𝑎2 𝑎3
𝑏1 𝑏2 𝑏3
𝑐1 𝑐2 𝑐3
| = 𝑎1 |
𝑏2 𝑏3
𝑐2 𝑐3
| − 𝑎2 |
𝑏1 𝑏3
𝑐1 𝑐3
| + 𝑎3 |
𝑏1 𝑏2
𝑐1 𝑐2
|
 
|
𝑎 𝑏
𝑐 𝑑
|
〈𝑎, 𝑏〉
〈𝑐, 𝑑〉
 |
𝑎1 𝑎2 𝑎3
𝑏1 𝑏2 𝑏3
𝑐1 𝑐2 𝑐3
|
〈𝑎1, 𝑎2, 𝑎3〉 〈𝑏1, 𝑏2, 𝑏3〉 〈𝑐1, 𝑐2, 𝑐3〉
https://www.youtube.com/watch?v=lYoWG9SNe14
http://tube.geogebra.org/material/simple/id/161530
https://www.geogebra.org/material/simple/id/844701
 
�̅� �̅� ℝ3
�̅� × �̅�
�̅� × �̅� = 〈𝑢1, 𝑢2, 𝑢3〉 × 〈𝑣1, 𝑣2, 𝑣3〉 = 〈𝑢2𝑣3 − 𝑣2𝑢3, 𝑢3𝑣1 − 𝑢1𝑣3, 𝑢1𝑣2 − 𝑢2𝑣1〉
〈𝑢2𝑣3 − 𝑣2𝑢3, 𝑢3𝑣1 − 𝑢1𝑣3, 𝑢1𝑣2 − 𝑢2𝑣1〉 = �̂� |
𝑢2 𝑢3
𝑣2 𝑣3
| − 𝒋̂ |
𝑢1 𝑢3
𝑣1 𝑣3
| + �̂� |
𝑢1 𝑢2
𝑣1 𝑣2
|
= |
�̂� 𝒋̂ �̂�
𝑢1 𝑢2 𝑢3
𝑣1 𝑣2 𝑣3
|
�̅� �̅�
‖�̅� × �̅�‖ = ‖�̅�‖‖�̅�‖|𝑠𝑒𝑛 𝜃|
𝑠𝑒𝑛2𝜃 = 1 − 𝑐𝑜𝑠2𝜃 = 1 −
 (
(𝑢∙�̅�)2
‖𝑢‖2‖�̅�‖2
)
http://www.youtube.com/watch?v=puy6BCjdDZU&index=11&list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-ihttp://www.youtube.com/watch?v=puy6BCjdDZU&index=11&list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i
http://www.youtube.com/watch?v=vMDskjMJ3F4
http://www.youtube.com/watch?v=vMDskjMJ3F4
http://www.youtube.com/watch?v=flxpZIYV-YM&index=12&list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i
 
�̅� = 〈𝑢1, 𝑢2, 𝑢3〉, �̅� =
〈𝑣1, 𝑣2, 𝑣3〉 y �̅�〈𝑤1, 𝑤2, 𝑤3〉 ℝ
3
[�̅�, �̅�, �̅�] = �̅� ∙ �̅� × �̅� = 𝑢1 |
𝑣2 𝑣3
𝑤2 𝑤3
| − 𝑢2 |
𝑣1 𝑣3
𝑤1 𝑤3
| + 𝑢3 |
𝑣1 𝑣2
𝑤1 𝑤2
| = |
𝑢1 𝑢2 𝑢3
𝑣1 𝑣2 𝑣3
𝑤1 𝑤2 𝑤3
|
�̅�, �̅� �̅�
https://www.youtube.com/watch?v=bb5Tk8EclzY&list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i&index=13
https://www.youtube.com/watch?v=bb5Tk8EclzY&list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i&index=13
http://www.youtube.com/watch?v=qqwQdnPh2x8&list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i&index=14
http://www.youtube.com/watch?v=eK5fIqqwyfg&index=15&list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i
http://www.youtube.com/watch?v=egTRnzrwCFk
http://www.youtube.com/watch?v=RnW7mH6jSEM
 
 
 
 
http://www.youtube.com/playlist?list=PLVEkI8DcwbMtXW1Ug8HklcTKLzoiDVn2F&spfreload=10
http://www.youtube.com/playlist?list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i&spfreload=10
http://www.youtube.com/playlist?list=PLVEkI8DcwbMs2AUWHRKg3dqcJM4-7I9-i&spfreload=10
http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010
http://creativecommons.org/licenses/by-nc-sa/4.0/
http://www.youtube.com/watch?v=wPQcA42XwGk