Linear Algebra

1. Define and apply the properties of vector spaces and subspaces.
2. Perform matrix operations and solve systems of linear equations.
3. Compute determinants and use them to find matrix inverses.
4. Determine eigenvalues and eigenvectors of matrices and linear transformations.
5. Diagonalize matrices and apply the Cayley-Hamilton theorem.

13 weeks, about 61 hours in total. Yours will differ.

1. Week 1Module 1: Vector Spaces

   • Unit 1: Vector Spaces · 4 hours

     Read the introduction to vector spaces.. Understand the definitions and examples of vector spaces.. Study spaces associated with vector spaces.. Define and understand vector subspaces..

2. Week 2Module 1: Vector Spaces

   • Unit 2: Linear Combinations · 4 hours

     Define linear combinations of column vectors.. Form linear combinations using different scalars.. Understand the consistency of a system.. Define and understand the linear span of a collection of vectors..

3. Week 3Module 1: Vector Spaces

   • Unit 3: Linear Transformation I · 5 hours

     Verify the linearity of mappings between vector spaces.. Construct linear transformations with specified properties.. Define the range and kernel of a linear transformation.. Calculate the rank and nullity of a linear operator..

4. Week 4Module 1: Vector Spaces

   • Unit 4: Linear Transformation II · 5 hours

     Prove and apply the Rank Nullity Theorem.. Define an isomorphism between two vector spaces.. Show that two vector spaces are isomorphic if and only if they have the same dimension.. Prove and use the fundamental theorem of homomorphism..

5. Week 5Module 2: Matrices

   • Unit 1: Matrices I · 5 hours

     Define and give examples of various types of matrices.. Obtain a matrix associated with a given linear transformation.. Define a linear transformation given its associated matrix.. Evaluate the sum, difference, product, and scalar multiples of matrices..

6. Week 6Module 2: Matrices

   • Unit 2: Matrices II · 5 hours

     Obtain the transpose and conjugate of a matrix.. Determine if a given matrix is invertible.. Obtain the inverse of a matrix.. Discuss the effect of change of basis on the matrix of a linear transformation..

7. Week 7Module 2: Matrices

   • Unit 3: Matrices III · 5 hours

     State the invertible matrix theorem.. State and prove the conditions for a matrix to be invertible.. Define and obtain the rank of a matrix.. Reduce a matrix to echelon form..

8. Week 8Module 3: Determinants

   • Unit 1: Determinants I · 4 hours

     Define the determinant of a matrix.. Evaluate the determinant of a square matrix using properties of determinants.. Obtain the minor, cofactors, and adjoint of a square matrix..

9. Week 9Module 3: Determinants

   • Unit 2: Determinants II · 4 hours

     Compute the inverse of an invertible matrix using its adjoint.. Apply Cramer's rule to solve systems of linear equations.. Understand the product formula for determinants.. Define and evaluate the determinant rank of a matrix..

10. Week 10Module 4: Eigenvalues and Eigenvectors

    • Unit 1: Eigenvalues and Eigenvectors · 5 hours

      Obtain the characteristic polynomial of a linear transformation or a matrix.. Obtain the eigenvalues, eigenvectors, and eigenspaces of a linear transformation or a matrix..

11. Week 11Module 4: Eigenvalues and Eigenvectors

    • Unit 1: Eigenvalues and Eigenvectors · 5 hours

      Obtain a basis of a vector space with respect to which the matrix of a linear transformation is in diagonal form.. Obtain a non-singular matrix P which diagonalizes a given diagonalizable matrix A..

12. Week 12Module 4: Eigenvalues and Eigenvectors

    • Unit 2: Characteristic and Minimal Polynomials · 5 hours

      State and prove the Cayley-Hamilton theorem.. Find the inverse of an invertible matrix using the Cayley-Hamilton theorem.. Prove that a scalar is an eigenvalue if and only if it is a root of the minimal polynomial..

13. Week 13Module 4: Eigenvalues and Eigenvectors

    • Unit 2: Characteristic and Minimal Polynomials · 5 hours

      Obtain the minimal polynomial of a matrix (or linear transformation) if the characteristic polynomial is known.. Review all modules and units.. Work on assignments and exercises..