# Codimension of the null space

• Mar 28th 2011, 04:18 AM
raed
Codimension of the null space
Dear Colleagues,

Let $\displaystyle f\neq0$ be any linear functional on a vector space $\displaystyle X$ show that in the quotient space $\displaystyle X/N(f)$ the codim$\displaystyle N(f)=1$. Here $\displaystyle N(f)$ denotes the null space of $\displaystyle f$, and codim means the dimension of $\displaystyle X/N(f)$.

Remark: two elements $\displaystyle x_{1}, x_{2}\in X$ belong to the same element of the quotient space $\displaystyle X/N(f)$ if and only if $\displaystyle f(x_{1})=f(x_{2})$.

Regards,

Raed.
• Mar 28th 2011, 04:38 AM
FernandoRevilla
Hint :

If $\displaystyle f\neq 0$ then, $\displaystyle \textrm{Im}f=\mathbb{K}$ i.e. $\displaystyle \dim (\textrm{Im}f)=1$.
• Mar 28th 2011, 05:07 AM
raed
I do not understand.
• Mar 28th 2011, 06:39 AM
FernandoRevilla
Quote:

Originally Posted by raed
I do not understand.

What does mean $\displaystyle f:X \to \mathbb{K}$ is different from $\displaystyle 0$ ?
• Apr 7th 2011, 06:21 PM
mr fantastic
Quote:

Originally Posted by raed
I do not understand.

Please be more specific. What part do you not understand?
• Apr 8th 2011, 05:06 AM
HallsofIvy
Do you understand what a linear functional is?

(I puzzled over FernandoRevilla's response until I realized I had read "functional" but was still thinking "transformation"!)