What Is the Dimension of Subspaces U and W in a Vector Space V?

[iamalexalright]

Homework Statement

$$V=R^{4}\ and\ a^{\rightarrow}, b^{\rightarrow}, c^{\rightarrow}, d^{\rightarrow}, e^{\rightarrow} \in V.$$

(I'll drop the vector signs for easier typing...)

$$a = (2,0,3,0), b = (2,1,0,0), c = (-2,0,3,0), d = (1,1,-2,-2), e = (3,1,-5,-2)$$

$$Let\ U \subseteq V be\ spanned\ by\ a\ and\ b.\ Let\ W \subseteq V\ be\ spanned\ by\ c,d,e$$

$$Compute\ dim_{F}U, dim_{F}W, dim_{F}(U \cap W)$$


2. The attempt at a solution

I guess start with the dimension. We know the vectors a and b span U and by inspection they are linearly independent. Now I'm confused, is the dimension 3 or 4? I think 4 because the vectors have four 'slots' but I also think 3 since the last 'slot' is zero for both.

Also, for $$U \cap W$$ I would have to prove that a,b,c,d,e are linearly independent before I can find the dimension, no?

 

[rochfor1]

What is the definition of basis?

 

[Mark44]

For dim U, the answer is neither 3 nor 4. You have two vectors that span U and are linearly independent. As rochfor1 asked, what is the definition of a basis?

 

[iamalexalright]

Yeah, I just realized my error, so for V and W it would be 2 (since c=d+e)