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## Main Question or Discussion Point

The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices.

I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix.

The answer given is here, relevant answer is (b):

Imgur link: http://i.imgur.com/DKwt8cN.png

I cannot understand how D is [itex]R^4[/itex], let alone the rest of the answer. I kind of get why there'd be orthogonal subspaces in that case, since it's diagonal...but that's just grasping at straws.

I can see how we might take the columns of D and form linear combinations from them, but those column vectors are in [itex]R^2[/itex]

I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix.

The answer given is here, relevant answer is (b):

Imgur link: http://i.imgur.com/DKwt8cN.png

I cannot understand how D is [itex]R^4[/itex], let alone the rest of the answer. I kind of get why there'd be orthogonal subspaces in that case, since it's diagonal...but that's just grasping at straws.

I can see how we might take the columns of D and form linear combinations from them, but those column vectors are in [itex]R^2[/itex]