Math 361, Spring 2016, Assignment 8
From cartan.math.umb.edu
Carefully define the following terms, then give one example and one non-example of each:
- (F-)linear combination (of elements of a vector space V with scalars from F).
- (F-)span (of a subset of a vector space).
- (F-)linear relation (among elements of V).
- (F-)linearly independent set.
- (F-)basis.
- Dimension (of a vector space over F).
- Dimension (of a field extension F→E).
- Degree (of a field extension).
- Degree (of an algebraic element).
Carefully state the following theorems (you do not need to prove them):
- Theorem concerning extension of linearly independent sets to bases.
- Theorem concerning refinement of spanning sets to bases.
- Theorem relating the cardinalities of two bases for the same vector space.
- Theorem concerning the orders of finite fields.
- Theorem relating algebraic elements to finite-dimensional subextensions.
- Dimension theorem.
Solve the following problems:
- Section 30, problems 1, 3, 5, 7, and 9.
- Let F denote the field Q(3√2), the smallest subfield of R containing Q and 3√2.
- (a) Compute the dimension of F when regarded as a vector space over Q.
- (b) Show that the polynomial x2−2 has no roots in F. (Hint: use the Dimension Theorem.)
- (c) Compute [Q(√2,3√2):Q].
- (d) Find an explicit basis for Q(√2,3√2), regarded as a vector space over Q. Then give several sample calculations in this field.
Questions:
- Under definitions, #7 seems to be an elaboration or special case of #6, and #8 seems to be just another name for the phenomenon resulting from the two. Am I mistaken, or is consolidation of the definitions acceptable, or would you rather have specific examples illustrating the situations in which each term would arise? *cough* -IB (talk) 14:03, 28 March 2016 (EDT)
- You're correct. #7 is a special case of #6, and #8 is identical to #7. No need for separate examples in #7 and #8 (though it would be nice to give an example for #6 that shows it's a more general concept than #7). -Steven.Jackson (talk) 14:19, 28 March 2016 (EDT)
Solutions:
Vocab
- 1. Suppose S⊆V. A linear combination of elements of S is a vector that can be written as the sum of scalar multiples of elements of S. That is, for vectors vi∈S, vector w can be written
- c1⋅v1+…+ci⋅vi for scalars ci if and only if w is a linear combination of those elements in S. ::* The zero vector is a linear combination of any nonempty subset of any vector space. Set all ci to the zero element of the field, and the sum becomes a sum of zero vectors, which is the zero vector. ::* If S is the singleton corresponding to the zero vector, any nonzero vector in V is not a linear combination of elements of S (Prove these - they seem a bit too obvious (or find better examples)) :2. The ''span'' of a subset S⊆V is the set of all possible linear combinations of elements of S. ::* ::* :3. A ''linear relation'' of a subset S⊆V is an ordered set of scalars which, when multiplied with their respective vectors in S and those resulting vectors are added, the sum is the zero vector. ::<small> N.B. The relation where all the scalars are the zero element is the ''trivial relation''</small> ::* ::* :4. If there are no possible linear relations in a subset of vectors of V, the members of that set are said to be linearly independent, and the set is a ''linearly independent set''. In other words, a set is ::linearly independent if there is no way to get the zero vector with any sums of scalar products of the elements of the set. ::* ::* :5. Keep the definitions of ''span'' and ''linear independent set'' in mind: a ''basis'' for V is a set which spans V and is a linearly independent set. ::* ::* :6. The cardinality of any basis of a vector space V is the ''dimension'' of V. The reason it is meaningful is because any two bases of a vector space have the same cardinality (see #3 in Theorems). ::* The dimension of R2 over R is 2, because a basis for R2 over R is {(0,1),(1,0)}, and the cardinality of that set is 2. ::* :7. If the vector space E is a field extension of some field F, then the ''dimension of the field extension'' F→E is the dimension of vector space E over F. This is often denoted [E:F] :8. That last term ("dimension of the field extension F→E") can be interchanged with the term ''the degree of the extension'' F→E. ::* Examples here would be redundant and are left as further exercise for the reader. :9. If α is algebraic over field F, then F(α) is a field extension of F and [F(α):F]≡deg(α,F) is the ''degree of'' α. ::* Q is a field, and Q(3√2) is isomorphic to Q[x]/<x3−2>, so [Q(3√2):Q]=3, so the degree of 3√2 is 3. ::* ==='"`UNIQ--h-6--QINU`"'Theorems=== Given a vector space V and S⊆V'''...''' :1. If S is a linearly independent set, it can be "grown" into a basis by adding linearly independent vectors until S spans. :2. If S is a spanning set, it can be "trimmed" into a basis by removing linearly dependent vectors from S until S is linearly independent, without removing the spanning property. :3. Given any two bases B and β of V, by theorems 1 and 2, B and β have the same cardinality. (This is important, because it means that the dimension of a vector space, which depends on this ::value, can only possibly be one number. You can't get conflicting dimensions for the same vector space.) <br> Given a finite field F'''...''' :4. The order of F must be a prime power. (''i.e.'' F must have q elements, where q=pn for some prime number p and natural number n) :5. If E is a field extension of F, and [E:F] <small>(see definition 7)</small> is finite, every element of E is algebraic over F. <br> :6. Given field F with extension E, and field K which is a subextension of E ''s.t.'' F≤K≤E'''...''' *E is a vector space over F, :K is a vector space over F, and :E is a vector space over K. *[E:F]=[E:K]⋅[K:F], where ⋅ is standard integer multiplication. (example: if F, K and E are isomorphic to each other, [E:F]=[E:K]⋅[K:F]=1⋅1=1 (the trivial case holds))