Is S Closed Under Addition in R^(2x2)?

  • Thread starter WTFsandwich
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In summary, the problem is to determine if the set S1 = {B \in R^{2x2} | AB = BA} is a subspace of R^{2x2}. The set is not empty as the 2 x 2 Identity matrix is contained in S1. To prove that S1 is a subspace, it must be shown that it is closed under addition and scalar multiplication. To show closure under addition, it is necessary to show that A(B + C) = AB + AC for any B and C in S1. Since vector multiplication is left-distributive, this can be rewritten as A(B + C) = AB + AC. Since B and C are both members of S1, AB
  • #1
WTFsandwich
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Homework Statement


Suppose A is a vector [tex]\in[/tex] [tex]R^{2x2}[/tex].

Find whether the following set is a subspace of [tex]R^{2x2}[/tex].

[tex]S_{1} = {B \in R^{2x2} | AB = BA}[/tex]


The Attempt at a Solution


I know that S isn't empty, because the 2 x 2 Identity matrix is contained in S.

The problem I'm having comes in the proof that addition is closed.

If I show A(B + C) = (B + C)A that should be sufficient, right?

So far I have:

Suppose [tex]B[/tex] and [tex]C[/tex] [tex]\in[/tex] S.
[tex]A(B + C) = (B + C)A[/tex]
[tex]AB + AC = BA + CA[/tex]

And that's where I'm stuck. I have no idea where to continue on to. Any help would be greatly appreciated.
 
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  • #2
You need to show that S is closed under addition and scalar multiplication. You probably want to do those separately.
 
  • #3
I know that's what I have to do, but I don't know how to go about doing it. I started the addition part up above, and am stuck at that point.
 
  • #4
OK, B and C are both elements of S.
A(B + C) = AB + AC (since vector multiplication is left-distributive)
Now, what can you say about AB and AC, since B and C are members of set S?
 

FAQ: Is S Closed Under Addition in R^(2x2)?

1. What is a subset?

A subset is a set that contains elements that are all part of a larger set. In other words, all of the elements in a subset are also present in the larger set.

2. What is R^(2x2)?

R^(2x2) is the set of all 2x2 matrices, which are rectangular arrays of numbers with two rows and two columns. In other words, it is the set of all possible 2x2 matrices.

3. How do you determine if a set is a subset of R^(2x2)?

In order to determine if a set is a subset of R^(2x2), you need to check if all of the elements in the set are also 2x2 matrices. If they are, then the set is a subset of R^(2x2).

4. What does it mean for S to be a subset of R^(2x2)?

If S is a subset of R^(2x2), it means that all of the elements in S are also 2x2 matrices. In other words, S is a smaller set that is contained within the larger set of 2x2 matrices.

5. Can a set be a subset of itself?

Yes, a set can be a subset of itself. This is because all of the elements in the set are also present in the larger set, which is itself. In other words, the set and the larger set are identical.

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