# Thread: need example, intersection R(T) is not equal to {0}

1. ## need example, intersection R(T) is not equal to {0}

Hi,
I would like some help with the following problem:
Find a vector space V and a linear transformation T: V .... V such that V = N(T) + R(T) and N(T) intersection R(T) is not equal to {0}
(V cant be finite dimensional)

Thank you.

2. Originally Posted by Nona
Hi,
I would like some help with the following problem:
Find a vector space V and a linear transformation T: V .... V such that V = N(T) + R(T) and N(T) intersection R(T) is not equal to {0}
(V cant be finite dimensional)

Thank you.
What precisely do you mean by N(T) and R(T)?

3. Originally Posted by Swlabr
What precisely do you mean by N(T) and R(T)?
Hi,
null space (kernel) N(T),
range (image) R(T).
Thank you

4. Originally Posted by Nona
Hi,
I would like some help with the following problem:
Find a vector space V and a linear transformation T: V .... V such that V = N(T) + R(T) and N(T) intersection R(T) is not equal to {0}
(V cant be finite dimensional)

Thank you.
$\displaystyle V:=\left\{ \left\{a_n\right\}_{n=1}^\infty \slash\,\,a_n \in \mathbb{R}\,,\,\,\forall n\in\mathbb{N}\right\}\,,\,\,T:V\rightarrow V\,,\,\,T\left\{a_n\right\}_{n=1}^\infty =\left\{a_2,0,0...\right\}$

$\displaystyle \mbox{Try now to find a very simple vector that is both in }N(T)\mbox{ and in }R(T)$

Tonio

5. Originally Posted by tonio
$\displaystyle V:=\left\{ \left\{a_n\right\}_{n=1}^\infty \slash\,\,a_n \in \mathbb{R}\,,\,\,\forall n\in\mathbb{N}\right\}\,,\,\,T:V\rightarrow V\,,\,\,T\left\{a_n\right\}_{n=1}^\infty =\left\{a_2,0,0...\right\}$

$\displaystyle \mbox{Try now to find a very simple vector that is both in }N(T)\mbox{ and in }R(T)$

Tonio
@Tonio - Haven't really followed your definition of V. Would you mind giving a quick example of a vector in V. Thanks

6. Originally Posted by tonio
$\displaystyle V:=\left\{ \left\{a_n\right\}_{n=1}^\infty \slash\,\,a_n \in \mathbb{R}\,,\,\,\forall n\in\mathbb{N}\right\}\,,\,\,T:V\rightarrow V\,,\,\,T\left\{a_n\right\}_{n=1}^\infty =\left\{a_2,0,0...\right\}$

$\displaystyle \mbox{Try now to find a very simple vector that is both in }N(T)\mbox{ and in }R(T)$

Tonio
Thank you very much.
I could not follow the definition.

7. Originally Posted by Nona
Thank you very much.
I could not follow the definition.

Hmmm...V is the real vec. space of all real sequences, with addition defined by $\displaystyle \left\{a_n\right\}+\left\{b_n\right\}=\left\{a_n+b _n\right\}$ and multiplcation by scalar defined by $\displaystyle k\cdot \left\{a_n\right\}=\left\{ka_n\right\}\,,\,\,with\ ,\,k \in \mathbb{R}$

This is fairly well-known nice example of infinite dimensional v.s. You can also focus on the v.s. of all convergent sequences, of all convergent to zero sequences, etc.

Tonio

8. Originally Posted by tonio
Hmmm...V is the real vec. space of all real sequences, with addition defined by $\displaystyle \left\{a_n\right\}+\left\{b_n\right\}=\left\{a_n+b _n\right\}$ and multiplcation by scalar defined by $\displaystyle k\cdot \left\{a_n\right\}=\left\{ka_n\right\}\,,\,\,with\ ,\,k \in \mathbb{R}$

This is fairly well-known nice example of infinite dimensional v.s. You can also focus on the v.s. of all convergent sequences, of all convergent to zero sequences, etc.

Tonio
Thank you very much! I appreciate your help!