Honors Linear Algebra and Differential Equations

Additional books:

Grading: There will be weekly homework assignments (25%), two midterms (25% each), and a final exam (25%).
GRADING SCALE:

A	A-	B+	B	B-	C+	C	C-	D+	D
90	87	83	80	77	73	70	67	63	60

COURSE PROGRAM:
1. Vector spaces (2 weeks).
      Linear dependence and independence. Subspaces. Bases and dimensions.
      Vector spaces associated with a matrix: Row space, Column space, Null space.
      Systems of linear equations. Gaussian elimination. Echelon form. Rank.
      Operations on vector spaces: sum, direct sum, quotient space, dual space.
2. Linear transformations (1 week).
      Matrices. Multiplication.Inverse transformation and its matrix.
      Elementary matrices and an algorithm of finding the inverse matrix.
      Change of basis formulas. Kernel and Image.
3. Ordinary differential equations (1 week).
      1-st order linear ODEs. 2-nd order linear ODEs with constant coefficients.
4. Determinants (2 weeks).
      Permutations: product, inverse, sign, diagrams. Expansion over permutations.
      Determinants of product, transpose, inverse. Row and column expansions.
      Inverse matrix. Cramer's rule. Van der Monde and Cauchy determinants.
      Laplace expansion. Binet--Cauchy formula. Resultant.
5. Linear operators (3 weeks).
      Eigenvalue and eigenvectors. Characteristic and minimal polynomials.
      Invariant subspaces. Cayley-Hamilton theorem. Jordan normal form theorem
      and its proof. Elementary divisors. Exponent of a matrix.
6. ODEs (2 weeks).
      Separable equations. Exact equations and (deRham) cohomology. Picard's
      successive approximation for the existence and uniqueness theorems. Systems
      of first order ODEs. Exponent of a matrix again. Autonomous sytems. Phase plane,
      phase portrait, singular points. Nodes, Saddles, Foci, Centers. Stability.
7. Bilinear forms and inner product (3 weeks).
      Matrices. Symmetric and skew symmetric forms. Positive definite forms and
      euclidean spaces. Length and angles. Couchy-Schwarz and triangle inequalities.
      Gram-Schmidt orthogonalization. Normal forms, signature, and Sylvester's inertia
      theorem. Gram matrix, distance to the subspace. Volume of a parallelepiped.
      Orthogonal and self-adjoint operators. Principal axis theorem. Hermitian spaces.
      Unitary and hermitian operators.
8. Multilinear algebra (1 week).
      Multilinear maps. Tensors. Tensor product of vector spaces. Tensors of type (p,q).
      Contraction.