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