MATH2040: Linear Algebra II
Finite-dimensional structural linear algebra, with precise field conditions, complete polynomial and Jordan proofs, worked examples and solved practice.
1. Spaces, maps and dimension
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Chapter 1
1. Spaces, maps and dimension
Use direct sums, kernels and images to reason about abstract vector spaces.
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1. Spaces, maps and dimension
Use direct sums, kernels and images to reason about abstract vector spaces.
Chapter 2
2. Coordinates and change of basis
Separate a linear map from the matrices representing it.
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2. Coordinates and change of basis
Separate a linear map from the matrices representing it.
Chapter 3
3. Eigenstructure and diagonalization
Connect eigenspaces, multiplicity and diagonalizability over the stated field.
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3. Eigenstructure and diagonalization
Connect eigenspaces, multiplicity and diagonalizability over the stated field.
Chapter 4
4. Inner products and orthogonal projection
Build orthonormal bases and nearest-point projections in finite dimensions.
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4. Inner products and orthogonal projection
Build orthonormal bases and nearest-point projections in finite dimensions.
Chapter 5
5. Adjoints and special operators
Use adjoints to recognize self-adjoint, normal and unitary operators.
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5. Adjoints and special operators
Use adjoints to recognize self-adjoint, normal and unitary operators.
Chapter 6
6. Spectral theorems
Prove and apply orthonormal diagonalization over the complex and real fields.
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6. Spectral theorems
Prove and apply orthonormal diagonalization over the complex and real fields.
Chapter 7
7. Optional supplement: quadratic forms
Classify real quadratic forms through orthogonal changes of variables.
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7. Optional supplement: quadratic forms
Classify real quadratic forms through orthogonal changes of variables.
Chapter 8
8. Optional supplement: Jordan form
Use generalized eigenvectors and nullities to understand defective operators.
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8. Optional supplement: Jordan form
Use generalized eigenvectors and nullities to understand defective operators.
Chapter 9
Cayley–Hamilton and minimal-polynomial applications: diagnosis and complete practice
Cayley–Hamilton and minimal-polynomial applications: diagnosis and complete practice
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Cayley–Hamilton and minimal-polynomial applications: diagnosis and complete practice
Cayley–Hamilton and minimal-polynomial applications: diagnosis and complete practice
Chapter 10
Constructing Jordan Bases from Kernel Filtrations
Constructing Jordan Bases from Kernel Filtrations
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Constructing Jordan Bases from Kernel Filtrations
Constructing Jordan Bases from Kernel Filtrations
Chapter 11
Computing Jordan Forms and Using Their Nilpotent Parts
Computing Jordan Forms and Using Their Nilpotent Parts
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Computing Jordan Forms and Using Their Nilpotent Parts
Computing Jordan Forms and Using Their Nilpotent Parts