# Hyperplane

• Mar 28th 2011, 04:29 AM
raed
Hyperplane
Dear Colleagues,

If $\displaystyle Y$ is a subspace of a vector space $\displaystyle X$ and codim$\displaystyle Y=1$, then every element of $\displaystyle X/Y$ is called a hyperplane parallel to $\displaystyle Y$. Show that for any linear functional $\displaystyle f\neq 0$ on $\displaystyle X$, the set $\displaystyle H_{1}=\{x\in X|f(x)=1\}$ is a hyperplane parallel to the null space $\displaystyle N(f)$ of $\displaystyle f$.

Regards,

Raed.
• Mar 28th 2011, 05:04 AM
tonio
Quote:

Originally Posted by raed
Dear Colleagues,

If $\displaystyle Y$ is a subspace of a vector space $\displaystyle X$ and codim$\displaystyle Y=1$, then every element of $\displaystyle X/Y$ is called a hyperplane parallel to $\displaystyle Y$. Show that for any linear functional $\displaystyle f\neq 0$ on $\displaystyle X$, the set $\displaystyle H_{1}=\{x\in X|f(x)=1\}$ is a hyperplane parallel to the null space $\displaystyle N(f)$ of $\displaystyle f$.

Regards,

Raed.

Hints:

S\Definition: A subspace $\displaystyle H$ of a vector space $\displaystyle V$ is called a hyperplane iff it is a maximal

subspace of it, i.e. $\displaystyle <H,v>=V\,,\,\,\forall v\in V$

Claim: a subspace of a vec. sp. is a hyperplane iff it is the kernel of a non-zero linear

functional on the space

Claim: a subspace of a finite dimensional v.s. is a hyperplane iff its dimension is one less

that then space's.

Tonio