1. Limited time only! Sign up for a free 30min personal 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!

Linear transformations and standart matrices

  1. Dec 6, 2011 #1
    1. The problem statement, all variables and given/known data
    Define the linear transformation [itex]T: R^{3} → R^{3}[/itex] by [itex]T(v)=[/itex] the projection of [itex]v[/itex] onto the vector [itex]w=(1,2,1)[/itex]

    Find the (standard matrix of [itex]T[/itex])


    2. Relevant equations
    [itex]T: V → W[/itex] is a function from V to W (which means that for each v in V, there is a T9v) in W such that:
    T(v+v')=T(v)+T(v'), T(cv) = cT(v) for all v,v'[itex]\in[/itex]V, c[itex]\in[/itex]F

    For T(v) = Av, the matrix A is called the standard matrix of T

    projection of v onto w = [itex](\frac{v \bullet w}{w \bullet w})(w)[/itex]

    3. The attempt at a solution
    I'm having problems understanding Linear Transformations at all, and I'm not really sure if this is at all correct ... I'm thinking I should apply T to [itex]e_1, e_2, and~ e_3[/itex].

    Because the transformation takes place in [itex]R^{3}[/itex], I know A will be a 3x3 matrix. Apply T to [itex]e_1[/itex], where proj [itex]e_1[/itex] onto w would be [itex](1/6, 2/6, 1/6)[/itex] and we can see that [itex]x=1/6, y=1/3, z=1/6[/itex], so should the first column of A be [itex] \begin{pmatrix} 1/6 \\ 1/3 \\ 1/6 \end{pmatrix}[/itex]? Then apply T to [itex]e_2[/itex], where proj [itex]e_2[/itex] onto w would be [itex](1/3, 2/3, 1/3)[/itex] and we can see that [itex]x=1/3, y=2/3, z=1/3[/itex], so the second column of A should be [itex] \begin{pmatrix} 1/3 \\ 2/3 \\ 1/3 \end{pmatrix}[/itex]. And again, applying T to [itex]e_3[/itex] proj [itex]e_3[/itex] onto w would be [itex](1/6, 2/6, 1/6)[/itex] and we can see that [itex]x=1/6, y=1/3, z=1/6[/itex], so should the third column of A be [itex] \begin{pmatrix} 1/6 \\ 1/3 \\ 1/6 \end{pmatrix}[/itex]?

    So should the matrix A be [itex] A=\begin{pmatrix} 1/6 & 1/3 & 1/6 \\ 1/3 & 2/3 & 1/6 \\ 1/6 & 1/3 & 1/6 \end{pmatrix}[/itex]? This seems odd to have the third row be the same as the first row and the third column the same as the third column.

    Is this correct, or am I completely off? Thanks!
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Linear transformations and standart matrices
  1. Linear Transformation (Replies: 0)

Loading...