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A is an m x n matrix with rank n. I'm trying to prove that I - AA$^\dagger$ is an orthogonal projection operator, and I believe I did that already. I just can't figure out what subspace this is on. I already know AA$^\dagger$ is in $\Bbb R ^m$.

To clarify, A$^\dagger$ is the pseudo inverse.

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    The only reasonable description is "the image (column space) of $A$". Do we know more about $A$? – 2017-02-11
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    @spaceisdarkgreen It doesn't specify. – 2017-02-11
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    To specify, A$^\dagger$ is the pseudo inverse – 2017-02-11
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    @user409800 Ahh, usually $\dagger$ represents adjoint and $+$ is pseudoinverse (you might add that to the problem). Makes more sense now. Deleting irrelevant comment. – 2017-02-11
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    @user409800 And yes, then $I-AA^+$ projects onto the orthogonal complement of the column space of $A$ – 2017-02-11

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