I have an assignment problem:

Which are subspaces of the set of all $2 \times 2$ matrices?

  1. All $2\times 2$ matrices with integer entries

  2. All $2 \times 2$ matrices where the entries sum to zero, i.e. $$a + b + c + d = 0$$

For $\#2$, here is my thinking:

Adding any two matrices with this property results in another matrix with the same property. So it's closed under matrix addition. It's also closed under scalar multiplication by any real number. Distributivity of all the different kinds works, too. The additive identity is just the zero matrix. The additive inverse is just the same as scalar-multiplying by $-1$.

I think it passes all of the vector space requirements. This means that it is a subspace.

For $\#1$, here is my thinking:

The set of $2\times 2$ matrices with real entries is a vector space over $\mathbb{R}$, which means that any subspace would have to be closed under scalar multiplication by real numbers.

However, the set of $2\times2$ matrices with integer entries is not closed under scalar multiplication by real numbers. For example, the matrix of entries all $1$ multiplied by the scalar $0.5$ results in a matrix with non-integer entries. So it fails closure under scalar multiplication, even though it passes "vector" addition and distributivity, additive inverses and identities, etc.

This means that it is not a subspace, I think. Is that correct?

Thanks for help.

  • 2
    $\begingroup$ Yep on both counts. The only quibble is that you'd like to make sure that your subspace isn't empty, but you've already stated that the zero matrix is in there, so that's fine. $\endgroup$
    – Randall
    Oct 22 '20 at 0:36
  • $\begingroup$ OK. Thank you . $\endgroup$
    – terrace
    Oct 22 '20 at 0:42

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