M1 Midterm 1 Review

Part 1: Groups

• (Lec. 1; Artin ch. 2) A group is a set with an operation such that
  1. there exists , the identity, such that for all ;
  2. for all there exists an inverse such that ;
  3. associativity: for all .
• A group is abelian if is commutative ( for all ).
• Examples: , , , the symmetric group , the dihedral group ; products with , e.g. , etc.
• (Lec. 2) Like sets, groups can be finite (, , …), countable (, , , …) or uncountable (, …).
• is a subgroup if ; ; . divides .
  • subgroups of is a subgroup of .
  • All subgroups of are for some .
• A homomorphism is a map such that for all ().
  • Isomorphism = bijective homomorphism; automorphism = isomorphism . is a group.
• The kernel of : , a subgroup of . injective .
  • The image of : , a subgroup of . surjective .
• (Lec. 3) Given , , , is a homomorphism with , the subgroup generated by .
  • , where = order of = .
  • Hence the cyclic group is if has order , and if has infinite order.
  • ( generate if every element of is a product of the and their inverses.)
• A subgroup determines an equivalence relation (axioms: ; ; and ): (Lec. 4) iff , whose equivalence classes are the (left) cosets .
  • The quotient set: . The index of : if finite.
• If is finite: subgroup divides ; ; prime .
• A subgroup is normal for all (left cosets = right cosets) for all (conjugate subgroups).
• The operation makes a group iff is a normal subgroup.
• For every homomorphism , is a normal subgroup of , and .
  • (Lec. 5) If is surjective, we have an exact sequence ( injective, surjective, ).
  • Examples: ( normal); ( iff ); .
  • A homomorphism factors through iff .
• is simple if its only normal subgroups are and itself. Examples: , prime; , .
• Example: the center is a normal subgroup (abelian: ).

Part 2: Linear algebra

• A field is a set with two operations: is an abelian group with identity ; is an abelian group with identity ; distributive law .
  • (Lec. 6) Examples: , , (characteristic : ); , prime (characteristic ).
• (Axler ch. 1) A vector space over is a set with addition ( an abelian group, ) and scalar multiplication , associative and distributive.
  • Examples: , , …
  • Subspace: closed under and .
• ; say span if .
  • (Axler ch. 2) Say are linearly independent if for all .
  • (Lec. 7) Basis = linearly independent vectors which span : then , , is an isomorphism.
• All bases of have the same cardinality .
  • Any linearly independent set can be completed to a basis.
• = linear maps : , . This is a vector space.
• (Axler ch. 3) Given bases of and of , represent by the column , and by the matrix whose columns represent in the basis : .
  • (Lec. 8) Then is represented in the basis by the column vector :
  • Change of basis: , i.e. ; then for , .
• is a direct sum decomposition if i.e. , , is an isomorphism.
• finite-dimensional iff and .
• Dim/rank formula: finite-dimensional, .
• There exist bases of , of such that where the first rows correspond to and the last columns to .
• (Lec. 9) Dual: .
  • basis of (finite-dimensional) dual basis of such that
  • The annihilator of is : .
  • The transpose of is , .
• Quotient: subspace is a vector space.
  • , , is surjective with kernel .
  • factors through (there exists with ) iff .

Linear operators

• (Axler ch. 5) is an invariant subspace for if .
  • Examples: , ; eigenspaces .
• (Lec. 10) If with invariant for , then there is a basis where is block diagonal: .
  • A basis of eigenvectors , , diagonalizable: .
• Eigenvectors of for distinct eigenvalues are linearly independent.
• (Lec. 11) If is algebraically closed (e.g. ) then any linear operator has an eigenvector.
  • Corollary: there exists a basis in which is upper triangular, .
  • is an eigenvalue of is not invertible appears on the diagonal in a triangular matrix representing .

Midterm 1 review problems

This is a fairly arbitrary selection of problems from the textbook that may be good for checking your understanding of the concepts. I also recommend going over the homework assignments (not worrying about any problems that were marked with a * or particularly tricky or lengthy).

• Axler chapter 2: 2A.12; 2B.8; 2C.8, 2C.15, 2C.19
• Axler chapter 3: 3B.3, 3B.5, 3B.19, 3B.22, 3B.24, 3B.25, 3B.27; 3C.2, 3C.7; 3D.5, 3D.10, 3D.17; 3E.1, 3E.15; 3F.20, 3F.22, 3F.24, 3F.25 (Note: Axler uses the notation for the field; “null” for “Ker” and “range” for “Im”; for ; for , for .)
• Axler chapter 5: 5A.7, 5A.13, 5A.24, 5A.33; 5B.10; 5C.10; 5D.3, 5D.4; 5E.3
• Artin chapter 2: 4.5, 4.7, 4.9, 4.10; 5.1, 5.2; 6.3, 6.10; 8.6, 8.7, 8.9, 8.10; 10.2; 11.1, 11.4; 12.1, 12.2; M.2, M.9, M.10 (A couple of these are a bit more involved than what is intended for problems on the midterm.)
• Artin chapter 3: 1.9; 3.1; 4.5; 5.3.
• Artin chapter 4: 4.2, 4.6.