Results 1 to 2 of 2

Math Help - Two questions concerning linear maps/

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    42

    Two questions concerning linear maps/

    1. Let V be a vector space of dimension n, and W be a vector space of dimension m <= n. Show that there exists a surjective linear map V -> W.

    2. Let id denote the identity map R^n -> R^n. Let B denote a basis B = {v1, v2, ..., vn} of R^n and let E denote the standard basis E. Compute the matrix representation of I with respect to B, E.

    It is obvious to me that a surjective linear map exists if W is of lower dimension than V. I am unsure of how exactly to show this, though. For the map f: V -> W to be surjective, an f(v) (such that v is an element of V) must exist for every w (w is an element of W).

    I'm unsure of what I'm attempting to do on the second problem.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Two questions concerning linear maps/

    Quote Originally Posted by pantsaregood View Post
    1. Let V be a vector space of dimension n, and W be a vector space of dimension m <= n. Show that there exists a surjective linear map V -> W.
    So think about this, a linear map T:V\to W is surjective if and only if there exists bases B,B' for V,W respectively such that T:B\to B' is a surjection. So, what if you started with a surjection \{x_1,\cdots,x_n\}\to\{y_1,\cdots,y_m\} (where these are the obvious bases from our problem) and........


    2. Let id denote the identity map R^n -> R^n. Let B denote a basis B = {v1, v2, ..., vn} of R^n and let E denote the standard basis E. Compute the matrix representation of I with respect to B, E.

    It is obvious to me that a surjective linear map exists if W is of lower dimension than V. I am unsure of how exactly to show this, though. For the map f: V -> W to be surjective, an f(v) (such that v is an element of V) must exist for every w (w is an element of W).

    I'm unsure of what I'm attempting to do on the second problem.
    Think about it this way, write v_j say as  (a_1,\cdots,a_n). Then, you know that \text{id}(v_j)=(a_1,\cots,a_n)=a_1e_1+\cdots+a_n e_n so that I can bet you may conclude that the j^{\text{th}} column of the matrix should just be v_j.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 10th 2010, 12:38 PM
  2. Linear Maps
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 2nd 2009, 12:14 AM
  3. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 27th 2008, 04:25 AM
  4. Linear maps. Proving its linear, and describing kernal.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 20th 2008, 01:46 AM
  5. linear maps
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: March 9th 2008, 10:01 PM

Search Tags


/mathhelpforum @mathhelpforum