Codimension of the null space

Dear Colleagues,

Could you please help me in the following problem:

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.

Hint :


If $\displaystyle f\neq 0$ then, $\displaystyle \textrm{Im}f=\mathbb{K}$ i.e. $\displaystyle \dim (\textrm{Im}f)=1$.

I do not understand.

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Originally Posted by raed

I do not understand.

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What does mean $\displaystyle f:X \to \mathbb{K}$ is different from $\displaystyle 0$ ?

Quote:

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Originally Posted by raed

I do not understand.

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Please be more specific. What part do you not understand?

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"!)