Introduction to Abstract Algebra
1. Groups.
• Definition. Abelian/Non-abelian.
• Basic properties. Order of an element, order of a group. Cyclic groups.
• Examples, Z ⊂ Q ⊂ R ⊂ C, Zn , n−th roots of unity, the circle S 1 , matrices.
2. Three important examples.
• Arithmetic in Zn .
• Symmetry group Sn . General structure: Even, odd permutations and cycles. Sign
and Alternating group An .
• Dihedral groups Dn . Rotations and reflections in Dn .
3. Subgroups
• Definition.
• The subgroup generated by {g1 , g2 , ...}.
• Cosets. Index [G : H]. Lagrange theorem.
• Quotients. Example Zn := Z/nZ.
4. Rings and Fields.
• Definition.
√
• Some examples. Z ⊂ Q ⊂ R ⊂ C, Zn , Z[i] := {a + bi|a, b ∈ Z}, Q[ 2] :=
√
{a + b 2|a, b ∈ Q}.
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Linear Algebra
1. Vector Spaces.
• Fields. Definition and properties.
• Definition of vector spaces over a field.
• Spanning, linear independence, sifting. Basis. Dimension. Coordinates with respect
to a basis.
• Examples. Rn , Cn . Lines, planes, ... defined by a set of vectors or by a system of
linear equations. Canonical basis. Geometric interpretation of linear structure.
• More examples. R[x]≤n :=Polynomials in x of degree at most n. Canonical basis.
• Subspaces. Dimension of the sum and intersection of two subspaces.
2. Linear transformations.
• Definition. Important example: Projection to a subspace.
• Kernel and Image. How to find their dimension and compute them. Rank-Nullity
Theorem.
• When a spanning and/or linear independent set is preserved?
• The vector space HomK (U, V ).
3. Matrices.
• K m,n := the vector space of m×n matrices over K. Basic properties, multiplication.
• Identification via a basis.
U and V
↔
vectors
↔
linear transformations ↔
composition
↔
Rn and Rm
Columns
n × m matrices
multiplication.
• Examples. Projection and Rotations.
• Effect of using a different basis. Changes of coordinates.
• Row and column reduction. Row reduced form of a matrix. Rank=Column rank =
row rank=determinantal rank.
• Application to solve linear systems. The augmented matrix.
• Determinant. Effect of column and row operations. Rules for calculating det(A).
The inverse of a matrix.
4. Eigenvalues and Eigenspaces of a linear map/matrix.
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