Results 1 to 3 of 3

Math Help - Is W a subspace of M22?

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    171

    Is W a subspace of M22?



    How exactly would I do this? I know for W to be a subspace of M22 it has to be a vector space under the operations of addition and scalar multiplcation, but how exactly would I show this? :S
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Is W a subspace of M22?

    Hint The zero matrix does not belong to W
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,397
    Thanks
    760

    Re: Is W a subspace of M22?

    if W was a subspace, then any scalar multiple of an element of W would have to be in W. in particular, -w has to be in W if w is. so if

    w=\begin{bmatrix}a&a+3\\0&b\end{bmatrix} we must have:

    \begin{bmatrix}-a&-a-3\\0&-b \end{bmatrix} \in W. here, we have a problem, because -a-3 isn't 3 more than -a.

    in fact, W fails all 3 tests for a subspace, it is not closed under matrix addition, or under scalar multiplication, nor does it contain the 0-matrix (choosing a= 0, or a=-3 doesn't work, we always get a non-zero entry).

    closure tests are important, although checking to see that 0 is an element is the "easy one" (a vector space has to have a vector sum identity).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subspace
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 20th 2011, 10:13 PM
  2. subspace of V
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 17th 2010, 03:10 PM
  3. Subspace spanned by subspace
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 9th 2010, 07:47 PM
  4. Subspace
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 9th 2010, 03:08 AM
  5. Subspace
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 2nd 2008, 03:34 AM

Search Tags


/mathhelpforum @mathhelpforum