1
vote
2answers
30 views

Show the following subspaces are invariant

Let $V$ be a vector space over a field $F$ and let $\alpha \in End(V)$. IF $W$ and $Y$ are subspaces of $V$ which are invariant under $\alpha$, show that both $W+Y$ and $W\cap Y$ are invariant under ...
1
vote
2answers
48 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
3
votes
1answer
22 views

Is there any example of usage for a vector space over the field of formal Laurent series?

The formal Laurent series over a field is a field. Is there any example where vector spaces over that field occur naturally?
1
vote
1answer
49 views

Is there a specific method to finding a basis for vector spaces over $\mathbb{Q}$ ?

I am stuck on the first one but there are 5 questions on this so I really need help with the process. If anyone can help with any of the following. i) Find a Basis for the field K = ...
0
votes
1answer
42 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
0
votes
3answers
81 views

Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V $of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
5
votes
0answers
97 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
2
votes
3answers
365 views

Finding a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$.

I have to find a basis for $\Bbb{Q}(\sqrt{2}+\sqrt{3})$ over $\Bbb{Q}$. I determined that $\sqrt{2}+\sqrt{3}$ satisfies the equation $(x^2-5)^2-24$ in $\Bbb{Q}$. Hence, the basis should be ...
1
vote
1answer
58 views

Set of Functions is a Vector Space problem

Let $F$ be a field. Consider the $F$-vector space $F(X, F)$ of all functions from $X = \{ 1, 2 \} \to F$. Define $e_1, e_2 \in F(X, F)$ by $e_1 = \{ (1, 1),(2, 0) \}$ and $e_2 = \{ (1, 0),(2, 1) \}$. ...
1
vote
1answer
140 views

Show C(X) is a vector space over $\mathbb R$ with the following operations?

I have a set of continuous functions, $C(X): X \rightarrow R$ on a compact metric space, and definitions of addition & multiplication: $$(f+g)(x) = f(x)+g(x)$$ $$(\lambda f)(x) = \lambda ...
1
vote
1answer
41 views

Calculations in $K$-Algebras

Suppose we have some field $K$ and non-zero elements $a,b,$ in $K$. Define $H=H(a,b)$ to be the $K$-algebra with basis $\{1,x,y,z \}$ over $K$ satisfying $$x^2=a, \\ y^2=b, \\ z=xy=-yx$$ Question: How ...
1
vote
1answer
48 views

What kind of field is denoted by $\mathbb{Z}_2^n$?

For a homework question I have to prove whether or not some set is a vector space over a field $\mathbb{Z}_2^n$, but I am not sure what this notation means and the textbook doesn't seem to clarify. ...
1
vote
1answer
48 views

Uniqueness of complements in general vector spaces

Motivation I asked the question: Is my proof correct: rank-nullity in a field $K$. Although the answer given by Marc van Leeuwen made perfect sense, it made me wonder about one thing: Introduction ...
1
vote
1answer
526 views

Prove that the field F is a vector space over itself.

How can I prove that a field F is a vector space over itself? Intuitively, it seems obvious because the definition of a field is nearly the same as that of a vector space, just with scalers instead of ...
1
vote
1answer
66 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
1
vote
0answers
35 views

Field structure of vectors in $\mathbb{R}^3$

Probably a trivial question: By representing vectors in $\mathbb{R}^2$ as complex numbers we can define multiplication of vectors so that $\mathbb{R}^2$ has a field structure. Can this be extended to ...
2
votes
1answer
108 views

Injection from an integral domain to its field of fractions.

I have a quick question about modules. Suppose that $R$ is an integral domain with field of fractions $K$. Then any free $R$-module is isomorphic to copies of direct sums of $R$, say $R^i$ . ...
1
vote
1answer
79 views

Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ with the following property?

Is there a subfield $K$ such that $\mathbb{Q} \subset K \subset \mathbb{R}$ (proper subset) as follows: $\mathbb{R}$ is a vector space over $K$ and has no finite generating set and $K$ is a vector ...
1
vote
1answer
37 views

Dependence of $|K|$ and the union of the subspaces.

If $V$ is a $K$-vector space with $|K|\ge n$ and $V_1, \ldots, V_n$ are subspaces of $V$ such that $V=V_1 \cup \cdots \cup V_n$, then $V=V_i$ for some $i$. Is there a counterexample that for $|K|< ...
3
votes
1answer
95 views

Linear Transformation over Subfield

Letting $F\subseteq K$ be fields, and $V$ a vector space over $K$. Certainly, $V$ is also a vector space over $F$. And if $\{e_1,...,e_n\}$ is a basis for $K$ over $F$ and ...
-1
votes
3answers
160 views

Possible Cardinality of a Field

The following question struck me as pretty interesting: Let $\Bbb F$ be a field of characteristic $p$ (a prime, of course). I'm then asked to show that $|\mathbb{F}| = p^n$ for some $n\geq 1$. ...
2
votes
2answers
80 views

Proving we have a basis for $F[x]$

So $F$ is an arbitrary field, and $F[x]$ denotes the set of of formal polynomials with coefficients in $F$. And $A=\{f_i \mid i\geq 1\}$. I need to show two things, If $A$ is such that $deg (f_i) ...
2
votes
1answer
272 views

Linearly dependent vectors over finite fields

My problem is as follows: Assume you have a vector space of dimension $(d + 1)$, with values over $GF(q)$. Every vector in this vector space can be regarded as an element of the extension field ...
0
votes
1answer
691 views

Basis for $\mathbb{Q}(\sqrt{2},\sqrt{3})$

How can I find a basis for the field extension $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$? I can show that [$\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]=4$ (so I am looking for 4 basis elements). ...
5
votes
1answer
752 views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...
2
votes
1answer
114 views

Vector Spaces v.s. Fields

Vector spaces seem to have a very similar definition to fields. Are vector spaces fields? Thanks.
5
votes
3answers
650 views

What's the Difference Between a Vector and an Hypercomplex Number?

What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties. Perhaps this question could be put more generally as: ...