به انگلیسی

Linear algebra with ring scalars

Mathematics · Abstract Algebra · ages 22-24

Name ______________________ Date ____________

Write (7, -4) = a(1, 0) + b(0, 1) in Z^2. What is a?
Answer: ______________
For the Z-module Z/2, which correctly describes the empty set?
- Independent but not spanning
- Dependent because it contains a zero vector
- Spanning because the module has two elements
- A basis because it has no dependence relation
Why does Z/2 have no basis as a module over Z?
- Its two elements force every spanning set to be infinite
- 2 kills every element, preventing a nonempty independent set
- The empty independent set spans both residue classes
Vector spaces have bases, but modules over a ring need not.
Circle one: True False
Every square complex matrix has Jordan form arising from its C[x]-module structure, where x acts through the matrix.
Circle one: True False
On C[x]/((x - 4)^3), multiplication by x has one Jordan block J3(4).
Circle one: True False
Why is C4 plus C2 not cyclic, despite having eight elements?
- Its size is 4 + 2 rather than 4 times 2
- Its identity has order 8 and cannot generate it
- Every element has order at most 4
- Every pair has infinite order over the integers
Which Z-module is isomorphic to Z/6?
- Z/2 plus Z/3
- Z/2 plus a free integer coordinate
- Z/3 plus a free integer coordinate
A 2 by 2 complex matrix has only eigenvalue 5 and an eigenspace of dimension 1. What is its Jordan form?
- [[5, 0], [0, 5]]
- [[0, -1], [1, 0]]
- [[5, 1], [0, 5]]
For the Z-module Z^2 plus Z/8, how many elements belong to its torsion submodule?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

7 · The combination equals (a, b), so the first coordinate gives a = 7.
Independent but not spanning · No nontrivial combination can use an empty set, but its span contains only zero.
2 kills every element, preventing a nonempty independent set · Every singleton has a relation with nonzero integer coefficient 2. The empty set is independent but does not span this nonzero module.
True · The Z-module Z/2 provides a nonzero module without an independent spanning set.
True · C[x] is a one-variable polynomial PID, and complex polynomial factors are linear.
True · The basis (x - 4)^2, x - 4, 1 gives the three-step Jordan chain.
Every element has order at most 4 · A pair's order is the lcm of coordinate orders, which divide 4 and 2. No pair can have order 8.
Z/2 plus Z/3 · The residue-pair map for coprime moduli 2 and 3 is a Z-module isomorphism.
[[5, 1], [0, 5]] · One eigenvector direction means one block, and total dimension 2 makes that block size 2.
8 · Only zero free coordinates can be annihilated by a nonzero integer. The remaining Z/8 coordinate has eight choices.