I am being asked this question:

Consider the vector space M(n × n, R) of n × n-matrices over R. Show that the subset of all diagonal matrices is a subvector space of M(n × n, R).

To my knowledge, a set is a subvector space if it satisfies 3 requirements.

1.) Zero exists in the set.

2.) The set is closed under addition

3.) The set is closed under scalar multiplication

I believe that zero exists in the set because all the elements in the matrix that aren't along the diagonal are zero. To my understanding, to prove that a set is closed under addition I must show that f(x+y) = f(x) + f(y) and to show that it is closed under scalar multiplication I must show that f(rx) = r $*$f(x). I do not know how to prove the last two conditions in regards to a matrix.

  • $\begingroup$ Just do it for $n=2$. Choose two diagonal matrices, A and B and choose scalars $\alpha$ and $\beta$ and show that $\alpha A + \beta B$ is a diagonal matrix. $\endgroup$ – akech Feb 25 '16 at 1:27
  • $\begingroup$ If you add two diagonal matrices what do you get ? What about when you multiply a diagonal matrix by a scalar ? $\endgroup$ – Quality Feb 25 '16 at 1:28
  • $\begingroup$ What you quote here are the axioms for a linear map. Not those for a subspace. $\endgroup$ – Friedrich Philipp Feb 25 '16 at 1:49
  • $\begingroup$ Your argument for the zero isn't sound. It's not that all elements along the are zero doesn't show that your subspace has a zero vector. It's the fact that the zero matrix is a diagonal matrix. Also, but not related to the question: @FriedrichPhilipp Hi, Fritz! $\endgroup$ – Roland Feb 25 '16 at 7:28
  • $\begingroup$ He is right. Hi Roland! 😊 $\endgroup$ – Friedrich Philipp Feb 25 '16 at 15:11

As FriedrichPhilipp said, the conditions you've written are for showing a function is a linear map, not for showing closure under vector addition and scalar multiplication.

You're right in your argument that $0 \in M_{n\times n}(\mathbb{R})$.

For closure under vector addition, you need to show if $x,y \in M_{n\times n}(\mathbb{R})$ (i.e. if $x$ & $y$ are $n\times n$ diagonal matrices) then $x+y$ is also a diagonal matrix.

For closure under scalar multiplication, you need to show if $k \in \mathbb{R}$ and $x \in M_{n\times n}(\mathbb{R})$ (i.e. if $k\in\mathbb{R}$ and $x$ is an $n\times n$ diagonal matrix) then $kx$ is also a diagonal matrix.

Now try writing out your arguments for these :-).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.