# Union of subspaces of a linear space

1. Mar 7, 2009

### de_brook

Is there a linear space V in which the union of any subspaces of V is a subspace except the trivial subspaces V and {0}? pls help

2. Mar 7, 2009

### yyat

A vector space V can only have non-trivial subspaces if $$\dim V\ge 2$$.
This means you can choose two linearly independent vectors u, w, which generate 1-dimensional subspaces U, W respectively. Can $$U\cup W$$ be a subspace? Hint: try to find a linear combination of u,w that is not in $$U\cup W$$.

3. Mar 7, 2009

### de_brook

I have tried searching for such spaces but i could only find for spaces whose dimension is less than 2.

4. Mar 7, 2009

### yyat

$$\dim\mathbb{R}^n=n$$, surely you knew that?

5. Mar 7, 2009

### de_brook

yyat; Do you mean i can obtain a linear subspace V of \mathbb{R}^n such that the union of any subspaces of V is a subspace of V?

6. Mar 7, 2009

8. Mar 7, 2009

### ThirstyDog

I think the short answer is No.

9. Mar 7, 2009

### de_brook

what about if you are not working n dimensional space, can you still find such a space

10. Mar 7, 2009

### de_brook

what about if you are not working with n-dimensional space, can you still find such a space?

11. Mar 7, 2009

### ThirstyDog

If the dimension of the space is less than two then the only subspace are V and {0} as yyat pointed out. Hence your question is answered in this case.

If the dimension of the space is greater or equal to two then consider spaces X and Y generated by linearly independent vectors x and y. x+y does not belong to $$X \Cup Y$$. Implying you can't pick any subspaces and the union will be a subspace.

12. Mar 7, 2009

### de_brook

thanks so much.

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