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Determinats. Determinant of a square matrix is nothing but the
(signed, higher dimensional) volume of the (higher dimensional) parallelepiped
spanned by the vectors represented by the columns (or rows) of the
matrix. ** We will see that it has lots of nice
properties, in particular, some sort of linearity, and learn how to calculate
it.
Systems of linear equations (definitions, related matrices, solving *, relation between homogeneous and non-homogeneous systems)
Matrices (basic concepts, matrix operations and their properties, elementary
transformations and elementary matrices, equivalence of matrices, echelon forms,
Gaussian elimination, nonsingular matrices and the inverse matrix *, criteria of invertibility)
Matrix transformations (basic concepts, examples)
Determinants: simple properties; poly-linearity; behavior under elementary
transformations; row and column expansion; evaluation of determinants**; relation to singularity of a matrix; inverse of a matrix;
application to solving linear systems; geometric applications
Vector spaces (basic definitions and properties; examples **)
Subspaces (definitions, properties, examples, detecting a subspace, defining
a subspace via homogeneous linear conditions vs. as a span, the solution space
of a homogeneous system, finding its basis = solving the system ***)
Linear combinations, linear independence, bases and isomorphisms (concepts,
detecting if a vector belongs to a given span and if a given set of vectors is
linearly independent***)
Bases and isomorphisms (concepts,
basis, coordinates, isomorphism to
R^{n}†, dimension, extending
a given set of vectors to a basis ***, finding a basis
in a given span ***)
Vector (sub-)spaces related to a matrix (row space, column space, null
space), their properties and relations, rank and nullity, existence/uniqueness
criteria for linear systems in terms of ranks
Inner products and inner product spaces (concept, properties, examples ††)
The matrix of an inner product; positive definite matrices
(if covered in class) †††
The Cauchy-Swarz inequality and its applications. Length (norm), distance,
angle. Triangle inequality
Orthogonality.
Properties of orthogonal (orthonormal) sets. Gram-Schmitd process
Orthogonal complement (notion, properties, finding). Orthogonal projection.
Distance between a vector and a subspace. Orthogonality of the subspaces related
to a matrix. "Quadratic"** minimization problems
Linear transformations: examples; kernel and range; matrix of a linear
transformation
Linear transformations (continued): change of bases; similarity
Diagonalization****; eigenvalues
and eigenvectors; characteristic polynomial and its invariance under similarity;
diagonalizability of symmetric matrices (self-adjoint operators)
Grading policy * I will take off a few
(2-3) points for arithmetical mistakes. However, a lot of
points will be taken off for `obvious' mistakes, i.e., either those that you can
easily avoid or those showing a deep misunderstanding of the subject. This
includes, but is not limited to, the following:
Wrong dimension in a physical problem
Things that don't make sense
Inverse matrix with a row/column full of "0"s
Basis with repeated vectors or zero vectors
(As it is impossible to predict every single mistake that the students
can make, I reserve
the right to decide what is `obvious' on the fly.)
Furthermore, solving a
different problem (other than stated), even if perfect, will give you
no credit* Of course, this only applies to problems that I am
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