Codimension of the null space

**raed** · March 28th 2011, 04:18 AM

Dear Colleagues,

Could you please help me in the following problem:

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

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

Regards,

Raed.

**FernandoRevilla** · March 28th 2011, 04:38 AM

Hint :

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

**raed** · March 28th 2011, 05:07 AM

I do not understand.

**FernandoRevilla** · March 28th 2011, 06:39 AM

Originally Posted by raed

I do not understand.

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

**mr fantastic** · April 7th 2011, 06:21 PM

Originally Posted by raed

I do not understand.

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

**HallsofIvy** · April 8th 2011, 05:06 AM

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"!)

Thread: Codimension of the null space

LinkBack

Thread Tools

Display

Codimension of the null space

Similar Math Help Forum Discussions

Question on null space/column space/row space of a matrix

Vector space and null space

Finding the basis for the column space and null space

[SOLVED] range space and null space

column space, null space

Search Tags