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Linear algebra, problem with linear maps and their matrices

  1. Nov 12, 2009 #1
    1. The problem statement, all variables and given/known data
    We have the following linear map:
    f:R^2-->R^3 f(x,y)=(x,3x+8y,x+y+11y)
    e is the standar basis of R^2 and
    a is the standar basis of R^3
    Question: Is there a linear map g: R^3-->R^2 that the matrix of (fog, a,a) is the 0 matrix?
    Is there a linear map g: R^3-->R^2 that the matrix of (gof, e,e) is the 0 matrix?




    2. Relevant equations



    3. The attempt at a solution

    First of all the matrix of f for the standar basis is:
    1 0
    Α= 3 8
    11 1
    I also know that in general the matrix of a fog (i dont know the word maybe junction or merge?) is the matrix of f multiplied with the matrix of g.
    Now, what i cant understand is how am I going to find what the matrix of fog or gof will look like, because f and g come from different dimension spaces. In other words if F is R^2-->R^3 and G is R^3-->R2 then what is FoG or GoF going to be? I prefer to get an answer to this rather than the actual homework question! Thanks in advance!

    P.S again sorry for bad english its quite hard for me to find all those weird terms.
     
  2. jcsd
  3. Nov 13, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Isn't this just f(x,y)= (x, 3x+8y, x+12y)?

    With "y+ 11y= 12y", the correct matrix is
    [tex]A= \begin{pmatrix}1 & 0 \\ 3 & 8 \\ 1 & 12\end{pmatrix}[/tex]

    [tex]\begin{pmatrix}1 & 0 \\ 3 & 8 \\ 11 & 1\end{pmatrix}[/tex]
    gives the linear transformation f(x,y)= (x, 3x+ 8y, 11x+ y).

    The word, in English, is "composition".

    A matrix representing a linear transformation from R3 to R2 must have two rows and three columns. It must be of the form
    [tex]B= \begin{pmatrix}a & b & c \\ d & e & f\end{pmatrix}[/tex]
    and
    [tex]BA= \begin{pmatrix}a & b & c \\ d & e & f\end{pmatrix}\begin{pmatrix}1 & 0 \\ 3 & 8 \\ 1 & 12\end{pmatrix}[/tex]

    A matrix representing a linear transformation from R3 to R2 must have three columns and two rows. It must be of the form
    [tex]B= \begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}[/tex]
    and
    [tex]AB= \begin{pmatrix}1 & 0 \\ 3 & 8 \\ 1 & 12\end{pmatrix}\begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix}[/tex]


    Except for the fact that the word "standard" has a "d" on the end, your English is excellent. Far better than my (put whatever language you please here).
     
    Last edited: Nov 14, 2009
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