Results 1 to 2 of 2

Math Help - Prove that W_1 and W_2 are independent: proof needs checking

  1. #1
    Member
    Joined
    Jan 2010
    Posts
    232

    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 V and W are vector spaces over a field F.

    Let W_1 and W_2 be subspaces of a finite-dimensional vector space V.
    Prove that W_1 and W_2 are independent if and only if 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).

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

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

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

    As such, W_1 and W_2 are independent.

    -----

    Tell me if my answer needs any clean-up or reordering.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    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 V and W are vector spaces over a field F.

    Let W_1 and W_2 be subspaces of a finite-dimensional vector space V.
    Prove that W_1 and W_2 are independent if and only if 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).

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

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

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

    As such, W_1 and W_2 are independent.

    -----

    Tell me if my answer needs any clean-up or reordering.
    .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linearly independent polynomials; answer not checking out.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 28th 2011, 11:11 PM
  2. prove linear independent
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 12th 2010, 04:38 PM
  3. Prove linearly independent
    Posted in the Advanced Algebra Forum
    Replies: 19
    Last Post: January 26th 2010, 05:25 AM
  4. Prove that Y1 and Y2 are independent
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: March 8th 2009, 09:43 PM
  5. Prove linearly independent
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 21st 2008, 11:02 PM

Search Tags


/mathhelpforum @mathhelpforum