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

  1. #1
    Newbie Nona's Avatar
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    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.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Nona View Post
    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)?
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  3. #3
    Newbie Nona's Avatar
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    Quote Originally Posted by Swlabr View Post
    What precisely do you mean by N(T) and R(T)?
    Hi,
    null space (kernel) N(T),
    range (image) R(T).
    Thank you
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  4. #4
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    Quote Originally Posted by Nona View Post
    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
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  5. #5
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    Quote Originally Posted by tonio View Post
    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
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  6. #6
    Newbie Nona's Avatar
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    Quote Originally Posted by tonio View Post
    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.
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  7. #7
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    Quote Originally Posted by Nona View Post
    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
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  8. #8
    Newbie Nona's Avatar
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    Quote Originally Posted by tonio View Post
    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!
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