1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Simple Linear algebra question but I have to have a brain freeze smh

  1. Feb 10, 2012 #1
    So I'm reading an example from my text on how to construct an isomorphism:

    Let V be in R^3 be defined by the single linear equation: x1 - x2 + x3 = 0 (those are suppose to be subscripts after the x's) and let W be in R^3 be the subspace defined by the single linear equation 2x1 + x2 - x3 = 0. Since dim(V) = dim(W) = 2, V and W are isomorphic. To construc an isomorphism we choose bases for V and W. A basis for v1 = (1,1,0) and v2 = (0,1,1). A basis for w1 = (1, -1, 1) and w2 = (-1/2, 1, 0)

    Question: How do I find the dim of V and W? Should be straight forward but I don't see it...smh, and how do they come up with those bases for V and W in the isomorphism?

    Thanks
     
  2. jcsd
  3. Feb 10, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi trap101! :smile:
    every linear equation reduces the degrees of freedom by 1

    start with 3, apply one equation, result: 2 :wink:

    (ie, in n dimensions, one equation gives you an n-1 space, two simultaneous equations give you an n-2 space, etc)
    just choose numbers as simple as possible that satisfy the equation (and are independent) …

    use 1s and 0s if possible, if not then try 2s … :wink:
     
  4. Feb 10, 2012 #3
    Shouldn't I be able to form a linear combination from the standard basis vectors and solve for the dim in a matrix as well?......That was the route I was taking, but it might be useful to remember that fact from now on.
     
  5. Feb 10, 2012 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    not following you :confused:
     
  6. Feb 10, 2012 #5
    this equation: x1 - x2 + x3 = 0 from the first part of my question, My intention was to write this out as a linear combination using the standard basis vectors in R^3, and then I wanted to find the basis from the reduced-echelon form of the matrix.

    Another question: How do those chosen bases relate to linear equations?
     
  7. Feb 10, 2012 #6

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    I'm surprised tiny-tim didn't say "(try using the X2 button just above the Reply box :wink:)"

    In terms of analytic geometry, the equations, x1 - x2 + x3 = 0 and 2x1 + x2 - x3 = 0 are equations of planes, so you can think of them as each having two dimensions.

    The vectors v1 = (1,1,0) and v2 = (0,1,1) each lie in the plane, V. Similarly, w1 = (1, -1, 1) and w2 = (-1/2, 1, 0) each lie in plane W.
     
    Last edited: Feb 10, 2012
  8. Feb 10, 2012 #7
    Another way you can look at it is this.
    The subspace you are talking about is this

    \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} with [itex] 2x_1 + x_2 - x_3 = 0[/itex]
    plugging in [itex] x_3 = 2x_1 + x_2[/itex]

    [itex]\begin{pmatrix} x_1 \\ x_2 \\ 2x_1 + x_2 \end{pmatrix} = x_1\begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix} + x_2\begin{pmatrix} 0 \\1 \\1 \end{pmatrix}[/itex]

    Those are a base for the subspace and so the dimension is 2.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Simple Linear algebra question but I have to have a brain freeze smh
  1. I have few questions (Replies: 12)

Loading...