Math 361, Spring 2015, Assignment 7
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Carefully define the following terms, then give one example and one non-example of each:[edit]
- Finitely generated extension.
- Vector space (over a field F).
- F-linear transformation (between two vector spaces).
- F-linear combination (of elements of a set S).
- F-span (of a set S).
- F-spanning set.
- F-linearly independent set.
- F-basis.
- dimFV.
- Coordinate vector (of →v∈V with respect to an ordered basis B⊆V).
- Matrix (of a linear transformation T:V→W with respect to a pair of ordered bases B⊆V and C⊆W).
Carefully state the following theorems (you do not need to prove them):[edit]
- Theorem concerning extension of linearly independent sets to bases.
- Theorem concerning refinement of spanning sets to bases.
- Theorem relating the cardinalities of various bases for the same vector space.
Solve the following problems:[edit]
- Section 30, problems 1, 3, 4, and 5.
- Suppose that Fι→E is a field extension. Show that the scalar multiplication a⋅e=ι(a)e turns E into a vector space over F. (Here the multiplication on the right hand side is carried out in the field E.)
- Regard C as an extension of R in the usual way. Calculate dimRC.
- Regard R as an extension of Q in the usual way. Calculate dimQR. (Warning: this is actually a fairly hard exercise in set theory. But at least see if you can guess the answer.)