Results 1 to 3 of 3

Math Help - Prove n x 1 vectors Linearly independent

  1. #1
    Junior Member
    Joined
    Sep 2012
    From
    sthlm
    Posts
    53

    Prove n x 1 vectors Linearly independent

    Let $A$ be an $n\times n$-matrix. Three column vectors $X_1,X_2,X_3$ are given, and we have that $$A^2X_1=A^2X_2=A^2X_3=0$$ We know that $AX_1,AX_2, AX_3$ are linearly independent. Prove that the column vectors $X_1,X_2,X_3,AX_1,AX_2,AX_3$ are also linearly independent.


    It is a hard problem for me, and my first thought was that $A^2X_1=0 \Rightarrow AX_1=0$. That is probably not correct and not leading to anything.


    I then concluded that $\det(AX_1,AX_2, AX_3)\not=0$ which can lead me somewhere, since it's a proof with linear independence and $\det\not=0 \Leftrightarrow$ *lin. independent*.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Sep 2012
    From
    sthlm
    Posts
    53

    Re: Prove n x 1 vectors Linearly independent

    Let me post this correctly

    Let A be an n\times n-matrix. Three column vectors X_1,X_2,X_3 are given, and we have that A^2X_1=A^2X_2=A^2X_3=0 We know that AX_1,AX_2, AX_3 are linearly independent. Prove that the column vectors X_1,X_2,X_3,AX_1,AX_2,AX_3 are also linearly independent.


    It is a hard problem for me, and my first thought was that A^2X_1=0 \Rightarrow AX_1=0. That is probably not correct and not leading to anything.


    I then concluded that \det(AX_1,AX_2, AX_3)\not=0 which can lead me somewhere, since it's a proof with linear independence and \det\not=0 \Leftrightarrow *lin. independent*.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,528
    Thanks
    773

    Re: Prove n x 1 vectors Linearly independent

    Quote Originally Posted by jacob93 View Post
    my first thought was that A^2X_1=0 \Rightarrow AX_1=0.
    This does not have to be the case.

    Quote Originally Posted by jacob93 View Post
    I then concluded that \det(AX_1,AX_2, AX_3)\not=0 which can lead me somewhere, since it's a proof with linear independence and \det\not=0 \Leftrightarrow *lin. independent*.
    AX_1,AX_2,AX_3 is not a square matrix unless n = 3, so the determinant is not defined.

    The statement is obvious if you try proving it by contradiction.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. are these vectors linearly independent?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 23rd 2012, 06:26 AM
  2. Linearly independent vectors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 8th 2011, 04:11 PM
  3. Linearly Independent Vectors
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 1st 2010, 03:22 PM
  4. three vectors are linearly independent
    Posted in the Algebra Forum
    Replies: 1
    Last Post: October 17th 2009, 07:20 PM
  5. Linearly Independent Vectors
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 9th 2008, 04:14 AM

Search Tags


/mathhelpforum @mathhelpforum