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Homework Help: 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


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    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##.
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