Vector Spaces Handwritten Notes Www.mathcity
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Vector Spaces: Handwritten notes by Atiq ur Rehman http://www.MathCity.org/atiq Partial Contents 1. 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 related theorems 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/msc It 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. If you have anything (notes, model paper, old paper etc.) to share with other peoples, you can send us to publish on MathCity.org. You may earn money by participating. For more information visit: http://www.MathCity.org/participate 1
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A handwritten notes of vector spaces, usually teaches in Pakistani universities. Best to prepare a Rings & Vector spaces section of Algebra paper in MSc (Mathematics).
Transcript of Vector Spaces Handwritten Notes Www.mathcity
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
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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
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********************************************** Available online at http://www.MathCity.org
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********************************************** Available online at http://www.MathCity.org
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********************************************** Available online at http://www.MathCity.org
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• 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
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********************************************** Available online at http://www.MathCity.org
**********************************************
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**********************************************
Available online at http://www.MathCity.org
**********************************************
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Vector Spaces: Handwritten notes
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• 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
**********************************************
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