Syllabus for the Qualifying Exam in Linear Algebra

  1. Fields:
    1. the field of rational numbers
    2. the field of real numbers
    3. the field of complex numbers
  2. Linear equations and matrices, reduction to row echelon form.
  3. Vector spaces:
    1. Vector spaces, subspaces, quotient spaces.
    2. Linearly independent sets.
    3. Linear transformations,
    4. kernel and image,
    5. projections (idempotent linear operators),
    6. the set of linear transformations between two vector spaces forms a vector space.
    7. Bases and dimension for finite dimensional vector spaces.
  4. Matrices and linear transformations between finite dimensional vector spaces:
    1. The matrix of a linear transformation with respect to a choice of bases,
    2. similarity of matrices and change of basis for linear transformations.
    3. The inverse of a matrix,
    4. the determinant of a square matrix,
    5. the characteristic polynomial,
    6. the minimal polynomial,
    7. eigenvectors,
    8. eigenvalues.
    9. Diagonalizability,
    10. Jordan canonical form for square matrices over the complex numbers.
    11. Cayley-Hamilton theorem.
    12. Rank + nullity = dimension of domain.
    13. Dual basis of dual vector space,
    14. dual of a linear transformation and transpose of a matrix.
  5. Finite dimensional inner product spaces:
    1. Symmetric bilinear forms,
    2. hermetian forms,
    3. non-degeneracy,
    4. positive definiteness,
    5. the matrix of a bilinear form.
    6. Self-adjoint, orthogonal and unitary transformations.
    7. The standard positive definite inner product on real n-space,
    8. length and angle,
    9. diagonalization of real symmetric matrices by unitary matrices.
    10. Orthogonal projection,
    11. Gram-Schmidt orthogonalization.

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