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Lectures on Linear Algebra
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I.
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n-Dimensional Spaces. Linear and Bilinear Forms |
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1. |
n-Dimensional vector spaces |
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2. |
Euclidean space |
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3. |
Orthogonal basis. Isomorphism of Euclidean spaces |
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4. |
Bilinear and quadratic forms |
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5. |
Reduction of a quadratic form to a sum of squares |
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6. |
Reduction of a quadratic form by means of a triangular transformation |
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7. |
The law of inertia |
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8. |
Complex n-dimensional space |
II. |
Linear Transformations |
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9. |
Linear transformations. Operations on linear transformations |
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10. |
Invariant subspaces. Eigenvalues and eigenvectors of a linear transformation |
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11. |
The adjoint of a linear transformation |
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12. |
Self-adjoint (Hermitian) transformations. Simultaneous reduction of a pair of quadratic forms to a sum of squares |
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13. |
Unitary transformations |
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14. |
Commutative linear transformations. Normal transformations |
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15. |
Decomposition of a linear transformation into a product of a unitary and self-adjoint transformation |
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16. |
Linear transformations on a real Euclidean space |
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17. |
External properties of eigenvalues |
III. |
The Canonical Form of an Arbitrary Linear Transformation |
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18. |
The canonical form of a linear transformation |
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19. |
Reduction to canonical form |
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20. |
Elementary divisors |
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21. |
Polynomial matrices |
IV. |
Introduction to Tensors |
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22. |
The dual space |
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23. |
Tensors |
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Dummy View - NOT TO BE DELETED