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Math Help - linear subspace

  1. #1
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    linear subspace

    Hello, hope you can help me to answer a question relatively short from one of my exercises about Vector spaces, I thank you in advance.

    So I have not managed to show simply that set E is a linear subspace of
    \mathbb{R}^4., to give a basis and the dimension :

    That is the statement:
    Let \{E=\{(a,b,b,a+b) \in \mathbb{R}^4;(a,b) \in \mathbb{R}^2\}.

    __________________________________________________ ________________
    What I 'd say :
    Applying the method to show that a set is a linear subspace :
    E\subset \mathbb{R}^4
    E \neq \emptyset
    To prove the stability of E for the addition and multiplication by a scalar, I considere 2 elements : "a" and "b" in E:

    \forall \lambda,\mu \in \mathbb{K} all that is necessary is to show that \lambda.a+\mu.b \in E
    __________________________________________________ _________

    In particular, I want you to help me to prove the stability of E.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by boolean View Post
    Hello, hope you can help me to answer a question relatively short from one of my exercises about Vector spaces, I thank you in advance.

    So I have not managed to show simply that set E is a linear subspace of
    \mathbb{R}^4., to give a basis and the dimension :

    That is the statement:
    Let \{E=\{(a,b,b,a+b) \in \mathbb{R}^4;(a,b) \in \mathbb{R}^2\}.

    __________________________________________________ ________________
    What I 'd say :
    Applying the method to show that a set is a linear subspace :
    E\subset \mathbb{R}^4
    E \neq \emptyset
    To prove the stability of E for the addition and multiplication by a scalar, I considere 2 elements : "a" and "b" in E:

    \forall \lambda,\mu \in \mathbb{K} all that is necessary is to show that \lambda.a+\mu.b \in E
    __________________________________________________ _________

    In particular, I want you to help me to prove the stability of E.
    Yes, you stated exactly what you need to do- now do it!
    You want a and b to be members of the set E:{(a, b, b, a+b)} for a and b: that is any set of four numbers where the second and third numbers are the same and the fourth number is the sum of the first two.

    So you can write u as (a, b, b, a+b) and v as (x, y, y, x+y).
    Go ahead and do the calculation [itex]\alpha u+ \beta v[/itex]. Is it of the same form as vectors in E?
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