# Projection onto a subspace

#### fredsmithsfc

The Projv(x) = A(ATA)-1ATx

I'm puzzled why this equation doesn't reduce to Projv(x) = IIx

since (ATA)-1 = A-1(AT)-1 so that should mean that A(ATA)-1AT = AA-1(AT)-1AT = II

What is wrong with my reasoning?

Thanks.

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#### Fredrik

Staff Emeritus
Gold Member
It doesn't look wrong to me. Where did you get the idea that the first expression is a projection operator?

#### monea83

The problem is that A will be rectangular (non-square) if you are projecting onto a subspace, and thus its inverse does not exist (e.g. A is a column vector for projection onto a line).

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#### fredsmithsfc

It doesn't look wrong to me. Where did you get the idea that the first expression is a projection operator?
Hi Fredrik,

I first saw it in the Khan Academy Linear Algebra video: "Lin Alg: A Projection onto a Subspace is a Linear Transformation" which is at this link: http://www.khanacademy.org/video/lin-alg--a-projection-onto-a-subspace-is-a-linear-transforma?playlist=Linear%20Algebra [Broken]

But I also found it at Wikipedia here: http://en.wikipedia.org/wiki/Projection_(linear_algebra)

and in the book "Matrix Analysis and Applied Linear Algebra" by Carl Meyer on page 430

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#### fredsmithsfc

The problem is that A will be rectangular (non-square) if you are projecting onto a subspace, and thus its inverse does not exist (e.g. A is a column vector for projection onto a line).
That makes sense. Thanks.