# Thread: Prove that P is a Subspace

1. ## Prove that P is a Subspace

Hi all,

Got asked the following question in lecture:

Let $\displaystyle \bf v$ $\displaystyle \in R^n$. Prove that $\displaystyle P = \{\bf u \in R^n: \bf v * \bf u = 0\}$ is a subspace of $\displaystyle R^n$.

I understand that I must show that:
1. P is not empty.
2. P is closed under addition
3. P is closed under scalar multiplication

Just not sure, how I would show the 2nd and 3rd property. Thanks

2. ## Re: Prove that P is a Subspace

Originally Posted by Oiler
Just not sure, how I would show the 2nd and 3rd property. Thanks
I suppose $\displaystyle *$ represents an inner product.

If $\displaystyle u_1,u_2\in P$ then, $\displaystyle v*(u_1+u_2)=v*u_1+v*u_2=0+0=0$ . Hence, $\displaystyle u_1+u_2\in P$ . Try the third one.

3. ## Re: Prove that P is a Subspace

U could also prove condition (2) and (3) together:
Let $\displaystyle u_1,u_2\in P$ and two scalars $\displaystyle k,l\in \matbb{R}$ then, $\displaystyle v* (ku_1+lu_2)=v* ku_1+v * lu_2=0$ and so $\displaystyle ku_1+lu_2 \in P$