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Thread: Prove that W_1 and W_2 are independent: proof needs checking

  1. #1
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    Prove that W_1 and W_2 are independent: proof needs checking

    I have a good deal of work done for this, but could use a second opinion (and could use some info on whether I'm missing anything).

    Suppose that $\displaystyle V$ and $\displaystyle W$ are vector spaces over a field $\displaystyle F$.

    Let $\displaystyle W_1$ and $\displaystyle W_2$ be subspaces of a finite-dimensional vector space $\displaystyle V$.
    Prove that $\displaystyle W_1$ and $\displaystyle W_2$ are independent if and only if $\displaystyle W_1\cap W_2=\{0\}$.

    Here is my work so far (well, it's how my Professor showed us during his office hours, so he probably made it incomplete).

    $\displaystyle \forall w_1\in W_1$ and $\displaystyle \forall w_2\in W_2$ such that $\displaystyle w_1+w_2=0$, then $\displaystyle w_1=w_2=0$
    If we suppose that $\displaystyle W_1,W_2$ are independent, this implies that $\displaystyle W_1\cap W_2=\{0\}$.
    Suppose that $\displaystyle w\in W_1\cap W_2$,
    Clearly, $\displaystyle w+(-w)=0$ (We can let $\displaystyle w_1=w$ and $\displaystyle w_2=-w$)
    $\displaystyle \Rightarrow w=0$

    Using the above, we now need to prove that $\displaystyle W_1\cap W_2=\{0\}\Rightarrow W_1,W_2$ are independent.
    Suppose $\displaystyle w_1\in W_1$ and $\displaystyle w_2\in W_2$ and that $\displaystyle w_1+w_2=0$.
    $\displaystyle \Rightarrow w_1=-w_2$ OR $\displaystyle w_2=-w_1$
    $\displaystyle \Rightarrow w_1\in W_2$ OR $\displaystyle w_2\in W_1$

    These imply that $\displaystyle w_1=w_2=0$, and that they are independent.

    As such, $\displaystyle W_1$ and $\displaystyle W_2$ are independent.

    -----

    Tell me if my answer needs any clean-up or reordering.
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  2. #2
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    Quote Originally Posted by Runty View Post
    I have a good deal of work done for this, but could use a second opinion (and could use some info on whether I'm missing anything).

    Suppose that $\displaystyle V$ and $\displaystyle W$ are vector spaces over a field $\displaystyle F$.

    Let $\displaystyle W_1$ and $\displaystyle W_2$ be subspaces of a finite-dimensional vector space $\displaystyle V$.
    Prove that $\displaystyle W_1$ and $\displaystyle W_2$ are independent if and only if $\displaystyle W_1\cap W_2=\{0\}$.


    What is for you "independent linear subspaces"??

    Tonio


    Here is my work so far (well, it's how my Professor showed us during his office hours, so he probably made it incomplete).

    $\displaystyle \forall w_1\in W_1$ and $\displaystyle \forall w_2\in W_2$ such that $\displaystyle w_1+w_2=0$, then $\displaystyle w_1=w_2=0$
    If we suppose that $\displaystyle W_1,W_2$ are independent, this implies that $\displaystyle W_1\cap W_2=\{0\}$.
    Suppose that $\displaystyle w\in W_1\cap W_2$,
    Clearly, $\displaystyle w+(-w)=0$ (We can let $\displaystyle w_1=w$ and $\displaystyle w_2=-w$)
    $\displaystyle \Rightarrow w=0$

    Using the above, we now need to prove that $\displaystyle W_1\cap W_2=\{0\}\Rightarrow W_1,W_2$ are independent.
    Suppose $\displaystyle w_1\in W_1$ and $\displaystyle w_2\in W_2$ and that $\displaystyle w_1+w_2=0$.
    $\displaystyle \Rightarrow w_1=-w_2$ OR $\displaystyle w_2=-w_1$
    $\displaystyle \Rightarrow w_1\in W_2$ OR $\displaystyle w_2\in W_1$

    These imply that $\displaystyle w_1=w_2=0$, and that they are independent.

    As such, $\displaystyle W_1$ and $\displaystyle W_2$ are independent.

    -----

    Tell me if my answer needs any clean-up or reordering.
    .
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