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I'm writing a linear algebra exam next week and it's come to my attention that the prof that designed the test uses a different convention for complex inner product than the one my prof taught me.

I learned that the complex inner product is linear in the second argument and conjugate linear in the first, ie: $$i \langle v, w\rangle = \langle v, iw\rangle = \langle -iv, w\rangle$$

The prof's version is linear in the first argument: $$i\langle v, w\rangle = \langle iv, w\rangle = \langle v, -iw\rangle$$

I understand it's just a matter of convention and that two vectors that are orthogonal under one convention are also orthogonal in the other.

But for the purposes of the exam, does this have any affect on formulae?

For example, projection of $v$ onto $w$ is $\langle v, w\rangle w/\langle w, w\rangle$ the way I learned it. If I'm using the prof's version do I need to flip it to get the answer they're expecting?

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The formula itself is the same, but if you compute it with your convention you'll get the conjugate of the expected answer. –  Arturo Magidin Dec 11 '11 at 20:01
    
In the example given, $<v,w>$ would be the complex conjugate of the expected $<v,w>$, but $<v,w>w$ would not. –  Robert Israel Dec 11 '11 at 21:42
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3 Answers 3

If you are going by the convention: $\langle \cdot, \cdot \rangle$ is linear in the first argument, then $\mathrm{proj}_w(v) = \frac{\langle v,w \rangle}{\langle w, w \rangle} w$ is the "correct" formula.

As Arturo pointed out in the comments the other formula gives you essentially the same answer:

$$ \mathrm{proj}_w(v) = \frac{\langle v,w \rangle}{\langle w, w \rangle} w = \frac{\langle v,w \rangle}{\langle w,v \rangle} \cdot \frac{\langle w,v \rangle}{\langle w, w \rangle} w $$

So the answers will differ by a scalar, $\langle v,w \rangle$ divided by its conjugate. However, as you mention, multiplying vectors by a scalar does not effect orthogonality.

One might say, "Who cares? Can't we just use either formula for the projection?" In some sense, yes. But notice that $\mathrm{proj}_w(v)$ as defined above is a linear operator (linear in $v$), whereas the other formula is linear in $w$ and conjugate linear in $v$ (which in my mind is a little weird).

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My advice would be to use the formulae you're used to, and (if you think the exam might be graded by someone other than your own prof) in any question where you think it might make a difference write a note saying that this is the convention you're using.

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The projection $$ {\rm proj}_w v $$ is supposed to be linear in $v$, so just make sure your formulas are also linear in $v$. If $\langle,\rangle$ is linear in the first argument then $\langle v,w\rangle w/\langle w,w\rangle$ is correct, while if $\langle,\rangle$ is linear in the second argument then use $\langle w,v\rangle w/\langle w,w\rangle$.

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