Results 1 to 2 of 2

Math Help - Column and Null space proof!

  1. #1
    Junior Member
    Joined
    Oct 2009
    Posts
    30

    Column and Null space proof!

    The question is :

    Let
    A be an mn matrix and B an nr matrix. If AB = O (the mr matrix of zeroes)
    prove that the column space of B is contained in the null space of A.

    Here is what I have gotten:

    We know:
    <br />
AB=0 <br />
Ax=0<br />

    Where the null space is the span of the set of all vectors x that satisy the equation. Let null space(A) = span( x_1,x_2,...,x_s)
    I let col. space(B)=span( c_1,c_2,...,c_k)

    Let v be in the col. space(B)

    Therefore v=a_1c_1+a_2c_2+...+a_kc_k where a_i are elements of R

    If we can show that v can be shown to be apart of the span( x_1,x_2,...,x_s) then we are done. I cant think of a way to go about doing this.

    Any help would be greatly appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,974
    Thanks
    1121
    Quote Originally Posted by joe909 View Post
    The question is :

    Let
    A be an mn matrix and B an nr matrix. If AB = O (the mr matrix of zeroes)
    prove that the column space of B is contained in the null space of A.

    Here is what I have gotten:

    We know:
    <br />
AB=0
    Ax=0<br />
    If presume you mean to say that if ABv= A(Bv)= 0, then, letting x= Bv, Ax= 0. Then you can stop there! If v is any r-dimensional vector, then ABv= 0 so that A(Bv)= 0. What ever Bv is, A(Bv)= 0. But the "column space" of B is simply the space of all vectors of the form Bv- the "image" of B. And the "null space of A" is simply the space of all vectors x such that Ax= 0. The fact that A(Bv)= 0 for all v says that every vector of the form Bv, that is, every vector in the column space of B, is in the null space of A.

    [FONT=Fn]Where the null space is the span of the set of all vectors x that satisy the equation.

    What equation?

    Let null space(A) = span( x_1,x_2,...,x_s)
    I let col. space(B)=span( c_1,c_2,...,c_k)

    Let v be in the col. space(B)

    Therefore v=a_1c_1+a_2c_2+...+a_kc_k where a_i are elements of R

    If we can show that v can be shown to be apart of the span( x_1,x_2,...,x_s) then we are done. I cant think of a way to go about doing this.

    Any help would be greatly appreciated!
    Talking about basis vectors is completely unnecessary.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Question on null space/column space/row space of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 1st 2011, 01:47 PM
  2. Column and Null Space?
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: November 9th 2011, 09:07 PM
  3. Replies: 1
    Last Post: January 14th 2011, 09:51 AM
  4. Orthogonal vectors & column and null space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 13th 2009, 07:07 AM
  5. column space, null space
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: August 31st 2008, 09:49 AM

Search Tags


/mathhelpforum @mathhelpforum