Results 1 to 7 of 7
Like Tree1Thanks
  • 1 Post By Gusbob

Math Help - Matrix invertibilty and equivalent statements

  1. #1
    Newbie
    Joined
    Mar 2013
    From
    Los Angeles
    Posts
    17
    Thanks
    11

    Question Matrix invertibilty and equivalent statements

    Hi!

    I never took linear algebra, and I'm studying it on my own for the sake of learning (MIT opencourseware, online books, etc).

    I'm currently working on that a matrix A is invertible, and I can show that an equivalent statement to this is that A is row equivalent to the n x n Identity matrix. Now, I'd like to show that the n x n identity matrix has n pivot positions. How would I get that started? I'm not looking for a formal proof, but an informal proof with justifications (doesn't have to be rigorous, just the sketch of the idea).

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Matrix invertibilty and equivalent statements

    The real reason is that the identity matrix is, by definition, in rref with a leading 1 on the diagonal and zeroes everywhere else. Therefore there is a pivot position in each column (and row), and there are n columns (and row). There are also some geometric interpretations, but I'm pretty sure it all boils down to this property at the end anyways.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2013
    From
    Los Angeles
    Posts
    17
    Thanks
    11

    Re: Matrix invertibilty and equivalent statements

    Thanks! That clarified it.

    My next one to clarify is to show the following are equivalent (keeping in mind that A is an invertible n X n matrix) :

    The columns of A span Rn.
    The columns of A form a basis of Rn.

    I'm still foggy on the idea of something "spanning" something else, as is the case of columns of A span Rn, and I don't know what it means that the columns of A form a basis of Rn. However, I do imagine that it is also true that the columns of A forming the basis of Rn implies that the columns of A span Rn. In other words, the first statement <==> the second statement.

    So how are these two statements logically equivalent to one another?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Matrix invertibilty and equivalent statements

    Are you familiar with the invertible matrix theorem?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Mar 2013
    From
    Los Angeles
    Posts
    17
    Thanks
    11

    Re: Matrix invertibilty and equivalent statements

    I'm familiarizing myself with it, but I noticed different texts show different paths to show the equivalence of those statements.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Mar 2013
    From
    Los Angeles
    Posts
    17
    Thanks
    11

    Re: Matrix invertibilty and equivalent statements

    If I state that the columns of A are independent and they form a span of Rn therefore, they form a basis for Rn, is that enough or am I missing more of a justification?

    I'm still foggy on the idea of span.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Jan 2008
    Posts
    588
    Thanks
    87

    Re: Matrix invertibilty and equivalent statements

    That is enough, assuming you know those two facts are true.

    For your original question:
    The columns of A span Rn.
    The columns of A form a basis of Rn.
    The following applies to n\times n matrices.

    Columns of A span \mathbb{R}^n \iff There are n pivot rows in A\iff There are n pivot columns in A \iff The columns of A form a linearly independent set. Note that the middle equivalence about pivot rows and columns is not true in general for m\times n matrices with m\not= n

    Using what you have said in the preceding post, the columns of A form a basis of \mathbb{R}^n. The reverse implication is trivial from the definition of basis.
    Thanks from semouey161
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Matrix True And False Statements
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 24th 2013, 06:31 PM
  2. Prove that the following statements are equivalent
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: November 25th 2011, 06:15 AM
  3. Prove these statements are equivalent
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 15th 2010, 04:46 PM
  4. Replies: 5
    Last Post: September 6th 2009, 03:53 PM
  5. Rings - Ideals - Equivalent statements?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 15th 2008, 11:34 AM

Search Tags


/mathhelpforum @mathhelpforum