1. ## Vector space

Let U be the set of all vectors in R3 that are perpendicular to a vector u. Prove that U is a vector space.

I have no idea where to even start. :S

2. It is enough to show that U is a subspace.
You need to check that if a,b in U and ka in U for every real number k
(i.e., need to check if you have two vectors that perpendicular to u, then the sum of these vectors still perpendicular to u and if you have a vector that perpendicular to u then the scalar multiple of the vector still perpendicular to u)

3. Originally Posted by lmschneider
Let U be the set of all vectors in R3 that are perpendicular to a vector u. Prove that U is a vector space.

I have no idea where to even start. :S
start with verifying that U has all the properties of a vector space. this is just going through a check list.

the problem is, what kind of creatures do we check? well, two vectors are perpendicular if their dot product is zero. so that gives you a way to characterize the vectors in U, and now we know what to perform our checklist with

4. Suppose the W & V are both vectors that are orthogonal to U and that s is a scalar.
What is $\left( {sW + V} \right) \cdot U$?
If it is zero then how does that prove subspace??