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!

Linear Algebra- Scalar Multiplication

  1. Apr 2, 2013 #1
    1. The problem statement, all variables and given/known data

    Let M2 denote the set of all 2x2 matrices. We define addition with the standard addition of matrices, but with scalar multiplication given by:

    k [itex]\otimes[/itex] [a b c d] = [ka b c kd] (note that they are matrices)


    Where k is a scalar. Which of the following fails to hold?

    a. m2 is closed under scalar multiplication
    b. (ks)matrix = k [itex]\otimes[/itex] ( s [itex]\otimes[/itex] (matrix)
    c. 1 [itex]\otimes[/itex] (matrix) = (matrix)
    d. k [itex]\otimes[/itex](matrix + matrix') = k (matrix) + k (matrix') [Too lazy to inpute the otimes]
    e. (k+s) (matrix) = k(matrix) + s(matrix)

    2. Relevant equations



    3. The attempt at a solution

    I think the answer is E because if you multiply k+s to the matrix first then you can't split them.
     
  2. jcsd
  3. Apr 2, 2013 #2

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    That's right. Note that all you have to do to prove that the statement
    $$\text{For all }A\in M_2(\mathbb R)\text{ and all }k,s\in\mathbb R\text{, we have }(k+s)\otimes A=k\otimes A+s\otimes A.$$ is false is to show that there's one choice of k,s,A such that ##(k+s)\otimes A\neq k\otimes A+s\otimes A##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Linear Algebra- Scalar Multiplication
Loading...