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

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October 21st 2009, 09:47 AM
Nona

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)

it's an extra credit Q please help.
Thank you.
October 21st 2009, 12:08 PM
Swlabr

Quote:

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)

it's an extra credit Q please help.
Thank you.

What precisely do you mean by N(T) and R(T)?
October 21st 2009, 12:13 PM
Nona

Quote:

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
October 21st 2009, 12:18 PM
tonio

Quote:

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)

it's an extra credit Q please help.
Thank you.

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

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

Tonio
October 21st 2009, 04:02 PM
aman_cc

Quote:

Originally Posted by tonio

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

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

Tonio

@Tonio - Haven't really followed your definition of V. Would you mind giving a quick example of a vector in V. Thanks
October 21st 2009, 04:14 PM
Nona

Quote:

Originally Posted by tonio

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

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

Tonio

Thank you very much.
I could not follow the definition.
October 21st 2009, 07:49 PM
tonio

Quote:

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 $\left\{a_n\right\}+\left\{b_n\right\}=\left\{a_n+b _n\right\}$ and multiplcation by scalar defined by $k\cdot \left\{a_n\right\}=\left\{ka_n\right\}\,,\,\,with\ ,\,k \in \mathbb{R}$

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
October 22nd 2009, 08:18 AM
Nona

Quote:

Originally Posted by tonio

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

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!