Results 1 to 5 of 5

Math Help - Subspaces

  1. #1
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782

    Subspaces

    Show that if dim \ W_1 \geq \ dim \ W_2 and dim(W_1+W_2)=1+dim(W_1 \cap W_2), then W_1+W_2=W_1 and W_1 \cap W_2=W_2.
    Proving W_1+W_2=W_1
    dim \ W_1 \geq \ dim \ W_2 \Rightarrow dim \ W_1-dim \ W_2 \geq 0 \Rightarrow \ dim \ W_1-dim \ W_2 \geq k \geq 0

    The value of k represents the dimension of a basis for the vectors in W_1 that are not in W_2.

    Hence W_2 \subset  W_1 (Was the proof above enough to justify this step?).

    Therefore \forall \ u \in W_1 and  \forall v \in W_2, u+v \in W_1+W_2 \subset W_1.

    Conversely, \underline{0} \in W_2 so W_1 \subset W_1+W_2.

    This gives that W_1+W_2=W_1.

    Proving W_1 \cap W_2=W_1
    dim(W_1 +W_2)=dim(W_1)+dim(W_2)-dim(W_1 \cap W_2)

    but we know that dim(W_1+W_2)=dim(W_1)

    dim(W_1)=dim(W_1)+dim(W_2)-dim(W_1 \cap W_2) \Rightarrow dim(W_1 \cap W_2)=dim(W_2) \Rightarrow \ W_1 \cap W_2=W_2

    (this is implied because they have the same dimension over the same vector space so they must be the same subspace).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Showcase_22 View Post

    dim \ W_1 \geq \ dim \ W_2 \Rightarrow dim \ W_1-dim \ W_2 \geq 0 \Rightarrow \ dim \ W_1-dim \ W_2 \geq k \geq 0

    The value of k represents the dimension of a basis for the vectors in W_1 that are not in W_2.

    Hence W_2 \subset W_1 (Was the proof above enough to justify this step?). No!

    Therefore \forall \ u \in W_1 and  \forall v \in W_2, u+v \in W_1+W_2 \subset W_1.

    Conversely, \underline{0} \in W_2 so W_1 \subset W_1+W_2.

    This gives that W_1+W_2=W_1.



    dim(W_1 +W_2)=dim(W_1)+dim(W_2)-dim(W_1 \cap W_2)

    but we know that dim(W_1+W_2)=dim(W_1)

    dim(W_1)=dim(W_1)+dim(W_2)-dim(W_1 \cap W_2) \Rightarrow dim(W_1 \cap W_2)=dim(W_2) \Rightarrow \ W_1 \cap W_2=W_2

    (this is implied because they have the same dimension over the same vector space so they must be the same subspace).
    if W_1 \subseteq W_2 or W_2 \subseteq W_1, then we're done. otherwise, W_1 \cap W_2 would be a proper subspace of both W_1 and  W_2. thus: \dim W_1 \geq 1 + \dim W_1 \cap W_2 and \dim W_2 \geq 1 + \dim W_1 \cap W_2.

    therefore: \dim (W_1+W_2)=\dim W_1 + \dim W_2 - \dim W_1 \cap W_2 \geq 2 + \dim W_1 \cap W_2, contradicting our assumption that  \dim(W_1 + W_2)=1 + \dim W_1 \cap W_2. Q.E.D.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Nacho's Avatar
    Joined
    Mar 2008
    From
    Santiago, Chile
    Posts
    135
    W_1 and W_2 are in the same vectorial space?

    a question: If V_1 and V_2 are in the same vectorial space, I can tell: <br />
\dim \left( {V_1 } \right) \leqslant \dim \left( {V_2 } \right) \Rightarrow V_1  \leqslant V_2 <br />
?

    where the second "mayor o equal" mean subspace
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Nacho View Post
    W_1 and W_2 are in the same vectorial space?
    of course! otherwise W_1 + W_2 would be undefined!


    a question: If V_1 and V_2 are in the same vectorial space, I can tell: <br />
\dim \left( {V_1 } \right) \leqslant \dim \left( {V_2 } \right) \Rightarrow V_1 \leqslant V_2 <br />
?

    where the second "mayor o equal" mean subspace
    not necessarily!
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by Nacho View Post
    W_1 and W_2 are in the same vectorial space?

    a question: If V_1 and V_2 are in the same vectorial space, I can tell: <br />
\dim \left( {V_1 } \right) \leqslant \dim \left( {V_2 } \right) \Rightarrow V_1  \leqslant V_2 <br />
?

    where the second "mayor o equal" mean subspace
    The command is \subseteq
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. subspaces...
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 28th 2011, 12:20 PM
  2. Replies: 0
    Last Post: October 13th 2009, 03:48 PM
  3. Subspaces
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 8th 2009, 09:18 AM
  4. Subspaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 26th 2009, 06:55 PM
  5. Subspaces
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 14th 2009, 05:50 PM

Search Tags


/mathhelpforum @mathhelpforum