Linked Questions

7
votes
3answers
3k views

A vector space over an infinite field is not a finite union of proper subspaces? [duplicate]

Show that if $V$ is a vector space over an infinite field $\mathbb{F}$, then $V$ cannot be written as set-theoretic union of a finite number of proper subspaces.
4
votes
2answers
583 views

Vector space over an infinite field which is a finite union of subspaces [duplicate]

Let $V=\bigcup_{i=1}^n W_i$ where $W_i$'s are subspaces of a vector space $V$ over an infinite field $F$. Show that $V=W_r$ for some $1 \leq r \leq n$. I know the result "Let $W_1 \cup W_2$ is a ...
1
vote
3answers
751 views

Prove that $\mathbb{R}^n$ cannot be a finite union of its hyperplanes . [duplicate]

Prove that $\mathbb{R}^n$ cannot be a finite union of its hyperplanes . I want a prove using linear algebra only and not functional analysis i tried by contradiction we know R^n is a vector space ...
1
vote
0answers
122 views

Vector space over $F_p$ not the union of $k\leq p$ subspaces [duplicate]

Possible Duplicate: If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$ The following problem is found in ...
2
votes
3answers
56 views

Let $V$ be a finite dimensional vector space over $R$, then is it possible to write $V$ as union of finitely many proper subspaces? [duplicate]

Let $V$ be a finite dimensional vector space over $R$, then is it possible to write $V$ as union of finitely many proper subspaces? I am not sure but suppose if we have $B$ as an ordered basis for ...
1
vote
1answer
56 views

Prove that $\mathbb R^n$ is not the union of finitely many its proper subspaces. [duplicate]

I am reading An Introduction to Algebraic Topology by Rotman. After proving Theorem 2.7: For every $k\geq 0$, euclidean space$\mathbb R^n$ contains $k$ points in general position, the book remarked:...
0
votes
0answers
32 views

Basic Linear Algebra problem about representing a vector space as its proper subspaces union [duplicate]

Suppose $V$ is a vector space over an infinte field $F$. Assume that $n<\infty$ and $W_i$ $(1<i<n \land i ,n\in\Bbb N)$ are proper subspaces of $V$.$(W_i\neq V \land W_i\neq \emptyset)$ ...
17
votes
6answers
4k views

A finite-dimensional vector space cannot be covered by finitely many proper subspaces?

Let $V$ be a finite-dimensional vector space, $V_i$ is a proper subspace of $V$ for every $1\leq i\leq m$ for some integer $m$. In my linear algebra text, I've seen a result that $V$ can never be ...
10
votes
3answers
4k views

Prove that the union of three subspaces of $V$ is a subspace iff one of the subspaces contains the other two.

Prove that the union of three subspaces of V is a subspace iff one of the subspaces contains the other two. I can do this problem when I am working in only two subspaces of $V$ but I don't know how ...
5
votes
2answers
199 views

Can a vector space over finite field be written as union of finite number of proper subspaces?

Recently, I solved a problem that says- If $V$ is a vector space over an infinite field. Prove that, V cannot be written as set-thoretic union of a finite number of proper subspaces. But is this ...
5
votes
2answers
194 views

A ring as a finite union of fields

Let a ring $R$ be a finite union of fields all having the same unit. I want to prove that $R$ is itself a field. I wrote $R=\bigcup _{i=0}^{n}F_i$, with $F_0=\{0,1\}$ and $F_i$'s are fields. Since we ...
5
votes
1answer
169 views

Proof of $X\cup Y\neq V$

Suppose $X,Y$ are subspaces of dimension $n-k$ of the vector space $V$ of dimension $n$. Why is it always true that $X\cup Y\neq V$?  I can show this by arguing that if $X=Y$ then clearly by the ...
6
votes
1answer
412 views

A linear operator $T:V\rightarrow V$ has a cyclic vector iff $f_T=m_T$ (minimal polynomial=characteristic polynomial) [duplicate]

I want to prove the following statement: Let linear operator $T:V\rightarrow V$ ($V$ is $n$-dimentional). Then there exists a vector $v$ such that $\left\{ v, Tv, ..., T^{n-1}v \right\}$ form a ...
0
votes
2answers
97 views

For a finite extension $K/F$ of degree $>1$, prove that $K$ is not the union of finitely many proper intermediate fields.

Let $K/F$ be a finite extension of degree $>1$. Prove that $K$ is not the union of finitely many proper intermediate fields. Since $K/F$ is a finite extension, we can write $K=F(a_1,a_2,\ldots,...
5
votes
1answer
165 views

Union of proper subspaces, which is correct?

My textbook asks this: Suppose that $K$ is a finite field with $k$ elements, and that $V$ is an $r$-dimensional vector space over $K$. Show that if $V = \bigcup_{i=1}^n U_i$, where $U_1,\dotsc,U_n$...

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