vector spaces: handwritten notes by atiq ur rehman http ...€¦ · 3. vector spaces, de nition and...
TRANSCRIPT
![Page 1: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/1.jpg)
Vector Spaces: Handwritten notes
by
Atiq ur Rehman
http://www.MathCity.org/atiq
Partial Contents1. Rings; definition and examples 1
2. Field, definition and examples 1
3. Vector spaces, definition and examples 2
4. Subspaces, definition and related theorems 3
5. Linear sum, definition and related theorems 4
6. Homomorphism, kernel, linear combination 6
7. Linear span, related theorem 7
8. Finite dimensional vector space, linear dependent and independent, related theorem 8
9. Basis of a vector space and related theorems 10
10. Quotient space and related theorems 15
11. Internal direct sum, external direct sum, vector space homomorphism and relatedtheorems 19
12. Hom(V,W) and related theorems 26
13. Dual spaces and related theorems 32
14. Null space, nullity and related theorems 34
15. Eigen value, eigen vector, examples and related theorems 34
16. Characteristic polynomial/equation/matrix, examples and related theorem 46
17. Calay Hamilton theorem 48
18. Minimum polynomial (or minimal polynomial) and related theorems 50
19. Similar matrices and related theorems 55
Available at http://www.MathCity.org/mscIt is highly recommended that DON’T use these notes as a reference.
If you have any question, ask at http://forum.mathcity.org
MathCity.org is a non-profit organization, working to promote mathematics in Pakistan. Ifyou have anything (notes, model paper, old paper etc.) to share with other peoples, youcan send us to publish on MathCity.org. You may earn money by participating. For moreinformation visit: http://www.MathCity.org/participate
1
![Page 2: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/2.jpg)
[1]
MathCity.org Merging man and maths
Vector Spaces (Handwritten notes) Written by: AWritten by: AWritten by: AWritten by: Atiq ur Rehman, Class: BS or MSc (Maths)tiq ur Rehman, Class: BS or MSc (Maths)tiq ur Rehman, Class: BS or MSc (Maths)tiq ur Rehman, Class: BS or MSc (Maths)
Available online at http://www.MathCity.org , Version: 1.0.0
![Page 3: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/3.jpg)
www.MathCity.org
[2]
Vector Spaces: Handwritten notes
![Page 4: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/4.jpg)
www.MathCity.org
[3]
Vector Spaces: Handwritten notes
![Page 5: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/5.jpg)
www.MathCity.org
[4]
Vector Spaces: Handwritten notes
********************************************** Available online at http://www.MathCity.org
**********************************************
![Page 6: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/6.jpg)
www.MathCity.org
[5]
Vector Spaces: Handwritten notes
![Page 7: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/7.jpg)
www.MathCity.org
[6]
Vector Spaces: Handwritten notes
![Page 8: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/8.jpg)
www.MathCity.org
[7]
Vector Spaces: Handwritten notes
********************************************** Available online at http://www.MathCity.org
**********************************************
![Page 9: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/9.jpg)
www.MathCity.org
[8]
Vector Spaces: Handwritten notes
********************************************** Available online at http://www.MathCity.org
**********************************************
• Home � http://www.mathcity.org
o FSc � http://www.mathcity.org/fsc
o BSc � http://www.mathcity.org/bsc
o MSc � http://www.mathcity.org/msc
o MPhil/PhD � http://www.mathcity.org/mphil-phd
o Old papers � http://www.mathcity.org/papers
o E-Books � http://www.mathcity.org/e-books
![Page 10: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/10.jpg)
www.MathCity.org
[9]
Vector Spaces: Handwritten notes
![Page 11: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/11.jpg)
www.MathCity.org
[10]
Vector Spaces: Handwritten notes
• Home � http://www.mathcity.org
o FSc � http://www.mathcity.org/fsc
o BSc � http://www.mathcity.org/bsc
o MSc � http://www.mathcity.org/msc
o MPhil/PhD � http://www.mathcity.org/mphil-phd
o Old papers � http://www.mathcity.org/papers
o E-Books � http://www.mathcity.org/e-books
![Page 12: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/12.jpg)
www.MathCity.org
[11]
Vector Spaces: Handwritten notes
![Page 13: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/13.jpg)
www.MathCity.org
[12]
Vector Spaces: Handwritten notes
********************************************** Available online at http://www.MathCity.org
**********************************************
![Page 14: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/14.jpg)
www.MathCity.org
[13]
Vector Spaces: Handwritten notes
![Page 15: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/15.jpg)
www.MathCity.org
[14]
Vector Spaces: Handwritten notes
**********************************************
Available online at http://www.MathCity.org
**********************************************
![Page 16: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/16.jpg)
www.MathCity.org
[15]
Vector Spaces: Handwritten notes
![Page 17: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/17.jpg)
www.MathCity.org
[16]
Vector Spaces: Handwritten notes
• Home � http://www.mathcity.org
o FSc � http://www.mathcity.org/fsc
o BSc � http://www.mathcity.org/bsc
o MSc � http://www.mathcity.org/msc
o MPhil/PhD � http://www.mathcity.org/mphil-phd
o Old papers � http://www.mathcity.org/papers
o E-Books � http://www.mathcity.org/e-books
**********************************************
Available online at http://www.MathCity.org
**********************************************
![Page 18: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/18.jpg)
www.MathCity.org
[17]
Vector Spaces: Handwritten notes
![Page 19: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/19.jpg)
www.MathCity.org
[18]
Vector Spaces: Handwritten notes
![Page 20: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/20.jpg)
www.MathCity.org
[19]
Vector Spaces: Handwritten notes
![Page 21: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/21.jpg)
www.MathCity.org
[20]
Vector Spaces: Handwritten notes
![Page 22: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/22.jpg)
www.MathCity.org
[21]
Vector Spaces: Handwritten notes
![Page 23: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/23.jpg)
www.MathCity.org
[22]
Vector Spaces: Handwritten notes
![Page 24: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/24.jpg)
www.MathCity.org
[23]
Vector Spaces: Handwritten notes
![Page 25: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/25.jpg)
www.MathCity.org
[24]
Vector Spaces: Handwritten notes
![Page 26: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/26.jpg)
www.MathCity.org
[25]
Vector Spaces: Handwritten notes
![Page 27: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/27.jpg)
www.MathCity.org
[26]
Vector Spaces: Handwritten notes
![Page 28: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/28.jpg)
www.MathCity.org
[27]
Vector Spaces: Handwritten notes
![Page 29: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/29.jpg)
www.MathCity.org
[28]
Vector Spaces: Handwritten notes
![Page 30: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/30.jpg)
www.MathCity.org
[29]
Vector Spaces: Handwritten notes
![Page 31: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/31.jpg)
www.MathCity.org
[30]
Vector Spaces: Handwritten notes
![Page 32: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/32.jpg)
www.MathCity.org
[31]
Vector Spaces: Handwritten notes
![Page 33: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/33.jpg)
www.MathCity.org
[32]
Vector Spaces: Handwritten notes
![Page 34: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/34.jpg)
www.MathCity.org
[33]
Vector Spaces: Handwritten notes
![Page 35: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/35.jpg)
www.MathCity.org
[34]
Vector Spaces: Handwritten notes
![Page 36: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/36.jpg)
www.MathCity.org
[35]
Vector Spaces: Handwritten notes
![Page 37: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/37.jpg)
www.MathCity.org
[36]
Vector Spaces: Handwritten notes
![Page 38: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/38.jpg)
www.MathCity.org
[37]
Vector Spaces: Handwritten notes
![Page 39: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/39.jpg)
www.MathCity.org
[38]
Vector Spaces: Handwritten notes
![Page 40: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/40.jpg)
www.MathCity.org
[39]
Vector Spaces: Handwritten notes
![Page 41: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/41.jpg)
www.MathCity.org
[40]
Vector Spaces: Handwritten notes
![Page 42: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/42.jpg)
www.MathCity.org
[41]
Vector Spaces: Handwritten notes
![Page 43: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/43.jpg)
www.MathCity.org
[42]
Vector Spaces: Handwritten notes
![Page 44: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/44.jpg)
www.MathCity.org
[43]
Vector Spaces: Handwritten notes
![Page 45: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/45.jpg)
www.MathCity.org
[44]
Vector Spaces: Handwritten notes
![Page 46: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/46.jpg)
www.MathCity.org
[45]
Vector Spaces: Handwritten notes
![Page 47: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/47.jpg)
www.MathCity.org
[46]
Vector Spaces: Handwritten notes
![Page 48: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/48.jpg)
www.MathCity.org
[47]
Vector Spaces: Handwritten notes
![Page 49: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/49.jpg)
www.MathCity.org
[48]
Vector Spaces: Handwritten notes
![Page 50: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/50.jpg)
www.MathCity.org
[49]
Vector Spaces: Handwritten notes
![Page 51: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/51.jpg)
www.MathCity.org
[50]
Vector Spaces: Handwritten notes
![Page 52: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/52.jpg)
www.MathCity.org
[51]
Vector Spaces: Handwritten notes
![Page 53: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/53.jpg)
www.MathCity.org
[52]
Vector Spaces: Handwritten notes
![Page 54: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/54.jpg)
www.MathCity.org
[53]
Vector Spaces: Handwritten notes
![Page 55: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/55.jpg)
www.MathCity.org
[54]
Vector Spaces: Handwritten notes
![Page 56: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/56.jpg)
www.MathCity.org
[55]
Vector Spaces: Handwritten notes
![Page 57: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/57.jpg)
www.MathCity.org
[56]
Vector Spaces: Handwritten notes
![Page 58: Vector Spaces: Handwritten notes by Atiq ur Rehman http ...€¦ · 3. Vector spaces, de nition and examples 2 4. Subspaces, de nition and related theorems 3 5. Linear sum, de nition](https://reader036.vdocuments.us/reader036/viewer/2022062318/5fdb944eaedefc0255568659/html5/thumbnails/58.jpg)
www.MathCity.org
[57]
Vector Spaces: Handwritten notes