Thinking about matrices/polynomials

  1. Hi,
    A homework problem I ran across awhile ago asked me to determine if a set of of 2x2 Matrices were a basis for the set of aa 2x2 matrices.

    Am I going to run into any pitfalls by thinking about such 2x2 matrices as vectors of 4 components? Basically what I did was turn each 2x2 matrix into a 4x1 vector. Each row represented an entry in matrix A (row 1 was A11, row 2 was A12, row 3 was A21, row 4 was A22).

    Basically, I have no problems in dealing with vectors but when I run across problems where I'm given either polynomials or matrices with columns I'm unsure as to how I can approach them. For polynomials I figure I can just treat each power of x as a seperate component of a vector. Any insight would be appreciate and sorry if this post is jibberish, I'm a little tired. Thanks!
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,312
    Staff Emeritus
    Science Advisor

    As long as you are dealing with matrices as a vector space you are not using matrix product so, yes, you can just think of 2x2 matrices as a 4 dimensional vector.
     
  4. Hurkyl

    Hurkyl 16,090
    Staff Emeritus
    Science Advisor
    Gold Member

    It sounds like you're just using the coordinates defined by the ordered basis:

    [tex]
    \left(
    \left[\begin{array}{ll}1 & 0 \\ 0 & 0 \end{array}\right]
    ,
    \left[\begin{array}{ll}0 & 1 \\ 0 & 0 \end{array}\right]
    ,
    \left[\begin{array}{ll}0 & 0 \\ 1 & 0 \end{array}\right]
    ,
    \left[\begin{array}{ll}0 & 0 \\ 0 & 1 \end{array}\right]
    \right)
    [/tex]

    and using coordinates is fine, though not always the most efficient method of working with vectors.


    (Yes, [tex]\left[\begin{array}{ll}1 & 0 \\ 0 & 0 \end{array}\right][/tex] is a vector, and so is [itex]x^3 - 4x + 17[/itex]. You sound like you might be confusing yourself by using "vector" as a synonym for "n-tuple")
     
    Last edited: Oct 31, 2005
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