Results 1 to 7 of 7

Math Help - Determining Linear Independence of this set

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    45

    Determining Linear Independence of this set

    x1(t) =
    [2-t]
    [t ]

    x2(t)=
    [t+1]
    [-2 ]

    x3(t)=
    [t]
    [t+2]

    These are supposed to be matrices, but I don't know how to format them correctly. Anyways, I want to determine whether or not this set of vector functions is linearly dependent. The way I know how to go about this would be to set up a matrix and then evaluate the determinant. I would then look at the coefficients of the determinant; if they add or subtract to zero, then it is linearly dependent. The problem is that these 3 functions would make for a 2x3 matrix and I can't evaluate the determinant of that one.

    Any suggestions? Thanks,

    Jim
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,184
    Thanks
    403
    Awards
    1
    Quote Originally Posted by Jim Newt View Post
    x1(t) =
    [2-t]
    [t ]

    x2(t)=
    [t+1]
    [-2 ]

    x3(t)=
    [t]
    [t+2]

    These are supposed to be matrices, but I don't know how to format them correctly. Anyways, I want to determine whether or not this set of vector functions is linearly dependent. The way I know how to go about this would be to set up a matrix and then evaluate the determinant. I would then look at the coefficients of the determinant; if they add or subtract to zero, then it is linearly dependent. The problem is that these 3 functions would make for a 2x3 matrix and I can't evaluate the determinant of that one.

    Any suggestions? Thanks,

    Jim
    If these are linearly dependent then it is possible to form one of them by a linear combination of the other two. Thus
    \left [ \begin{matrix} 2 - t \\ t \end{matrix} \right ] = a \cdot \left [ \begin{matrix} t + 1 \\ -2 \end{matrix} \right ] + b \cdot \left [ \begin{matrix} t \\ t + 2 \end{matrix} \right ]

    See if you can find constants a and b such that this is true. If you can't, then this set is linearly independent.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2008
    Posts
    45
    Hey thanks for the suggestion. Could you give me an example as to how you go about finding these constants? Thanks,

    Jim
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by Jim Newt View Post
    Hey thanks for the suggestion. Could you give me an example as to how you go about finding these constants? Thanks,

    Jim
    I dont know why you are posting this again when Reckoner has worked out the answer for you here a few days ago.If you had any clarification, you could have asked it in the same thread. This is clearly a case of double posting... but over days.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Feb 2008
    Posts
    45
    Sorry about that, last time I looked at that thread, I didn't see that he had worked it out. Thanks!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by Jim Newt View Post
    Sorry about that, last time I looked at that thread, I didn't see that he had worked it out. Thanks!
    Never mind. If you didnt understand anything just ask in the same thread. That way people who answer feel appreciated. I am sorry if I sounded rude.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,184
    Thanks
    403
    Awards
    1
    Quote Originally Posted by Jim Newt View Post
    Hey thanks for the suggestion. Could you give me an example as to how you go about finding these constants? Thanks,

    Jim
    Just solve the system for any possible values:
    \left [ \begin{matrix} 2 - t \\ t \end{matrix} \right ] = a \cdot \left [ \begin{matrix} t + 1 \\ -2 \end{matrix} \right ] + b \cdot \left [ \begin{matrix} t \\ t + 2 \end{matrix} \right ]

    = \left [ \begin{matrix} a(t + 1) + bt \\ -2a + b(t + 2) \end{matrix} \right ]

    So we need to solve
    2 - t = a(t + 1) + bt
    and
    t = -2a + b(t + 2)
    for a and b in terms of t.

    Solve the top equation for b
    b = \frac{2 - t + at + a}{t}

    and insert this into the bottom equation:
    t = -2a + \left ( <br />
\frac{2 - t + at + a}{t} \right ) (t + 2)

    Solve this for a and substitute back to find b.

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear Algebra: Linear Independence question
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 3rd 2011, 06:28 AM
  2. Determining Linear Independence
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 9th 2011, 07:09 PM
  3. Replies: 5
    Last Post: December 15th 2010, 01:32 AM
  4. Replies: 4
    Last Post: September 28th 2010, 02:35 AM
  5. Determining Linear Independence
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 26th 2009, 10:40 AM

Search Tags


/mathhelpforum @mathhelpforum