Vector Spaces notes - Linear Algebra PDF

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Vector Spaces notes - Linear Algebra...

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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

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Vec t or S p a c es (H a n d w r i t t en n o t e s )       

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Vector Spaces: Handwritten notes



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

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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



********************************************** Available online at http://www.MathCity.org **********************************************

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

[5]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[6]

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Vector Spaces: Handwritten notes

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[7]

Vector Spaces: Handwritten notes



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[8]



[9]

Vector Spaces: Handwritten notes

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

[10]



[11]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



********************************************** Available online at http://www.MathCity.org **********************************************

[12]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



********************************************** Available online at http://www.MathCity.org **********************************************

[14]



[15]

Vector Spaces: Handwritten notes

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

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[16]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



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

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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes

 

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

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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



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

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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[24]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[26]



[27]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[28]



[29]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[30]



[31]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[32]



[33]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[34]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[36]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[38]



[39]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[40]



[41]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[42]



[43]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[44]



[45]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[46]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[48]



[49]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[50]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[52]



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Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[54]



[55]

Vector Spaces: Handwritten notes

Vector Spaces: Handwritten notes



[56]



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Vector Spaces: Handwritten notes...