1. ## Hyperplane

Dear Colleagues,

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

Regards,

Raed.

2. Originally Posted by raed
Dear Colleagues,

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

Regards,

Raed.

Hints:

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

subspace of it, i.e. $=V\,,\,\,\forall v\in V$

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