Linear Algebra- Scalar Multiplication

[FinalStand]

Homework Statement

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 \otimes [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 \otimes ( s \otimes (matrix)
c. 1 \otimes (matrix) = (matrix)
d. k \otimes(matrix + matrix') = k (matrix) + k (matrix') [Too lazy to inpute the otimes]
e. (k+s) (matrix) = k(matrix) + s(matrix)

Homework Equations

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.

 

[Fredrik]

Homework Statement

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 \otimes [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 \otimes ( s \otimes (matrix)
c. 1 \otimes (matrix) = (matrix)
d. k \otimes(matrix + matrix') = k (matrix) + k (matrix') [Too lazy to inpute the otimes]
e. (k+s) (matrix) = k(matrix) + s(matrix)

Homework Equations

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.

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 $(k+s)\otimes A\neq k\otimes A+s\otimes A$.