# 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 $V$.$(W_i\neq V \land W_i\neq \emptyset)$

Prove that we can't represent $V$ this way:

$V=\cup_1^n W_i=W_1\cup W_2\cup...\cup W_n$

I am spending my first course in linear algebra so i am not familiar with common concepts such as dimension or basis or... I am just starting this course and I only understand concepts like vector spaces or subspaces or fields or... please help me to solve this problem

thank you...

• But one actually can represent a vector space as a finite union of its subspaces when it’s finite, and it could be. So—at least in full generality—your claim is invalid. – arseniiv Dec 13 '17 at 23:51
• @arseniiv You are right... But I didn't write my question properly so I edit it... please check it again... thank you for your consideration... – Senmorta Dec 14 '17 at 10:44
• OK. But now this question has been marked as a duplicate of another; could you please elaborate what’s in answers to that one is still unclear? – arseniiv Dec 14 '17 at 14:30
• Sure. In answers you can see concepts are used that I am not familiar with which are not necessary to solve the problem according to my textbook. – Senmorta Dec 14 '17 at 16:41
• I had just reread answers there, and I think there isn’t a point trying to pick out dimension arguments, as even the resulting proof –not to mean translation process– would be tedious and uninspiring. An unlucky situation. – arseniiv Dec 14 '17 at 22:17