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I was wondering if a dot product is technically a term used when discussing the product of $2$ vectors is equal to $0$. And would anyone agree that an inner product is a term used when discussing the integral of the product of $2$ functions is equal to $0$? Or is there no difference at all between a dot product and an inner product?

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In my experience, the dot product refers to the product $\sum a_ib_i$ for two vectors $a,b\in \Bbb R^n$, and that "inner product" refers to a more general class of things. (I should also note that the real dot product is extended to a complex dot product using the complex conjugate: $\sum a_i\overline{b}_i)$.

The definition of "inner product" that I'm used to is a type of biadditive form from $V\times V\to F$ where $V$ is an $F$ vector space.

In the context of $\Bbb R$ vector spaces, the biadditive form is usually taken to be symmetric and $\Bbb R$ linear in both coordinates, and in the context of $\Bbb C$ vector spaces, it is taken to be Hermetian-symmetric (that is, reversing the order of the product results in the complex conjugate) and $\Bbb C$ linear in the first coordinate.

Inner products in general can be defined even on infinite dimensional vector spaces. The integral example is a good example of that.

The real dot product is just a special case of an inner product. In fact it's even positive definite, but general inner products need not be so. The modified dot product for complex spaces also has this positive definite property, and has the Hermitian-symmetric I mentioned above.

Inner products are generalized by linear forms. I think I've seen some authors use "inner product" to apply to these as well, but a lot of the time I know authors stick to $\Bbb R$ and $\Bbb C$ and require positive definiteness as an axiom. General bilinear forms allow for indefinite forms and even degenerate vectors (ones with "length zero"). The naive version of dot product $\sum a_ib_i$ still works over any field $\Bbb F$. Another thing to keep in mind is that in a lot of fields the notion of "positive definite" doesn't make any sense, so that may disappear.

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    $\begingroup$ I am scared by your answer (+1) :) It starts like mine, and viceversa! $\endgroup$ – Avitus Aug 26 '13 at 19:19
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    $\begingroup$ Actually in $\mathbb{C}^n$, the standard inner product is $$\sum_{i=1}^n a_i\bar{b_i}$$ note the complex conjugate. $\endgroup$ – AlexR Aug 26 '13 at 19:20
  • $\begingroup$ @AlexR Yeah, I was striving to find a logical location to introduce it. I would still call the other one a dot product though... although it's not nearly as useful as the positive definite version, of course :) $\endgroup$ – rschwieb Aug 26 '13 at 19:31
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    $\begingroup$ @rschwieb In our linear algebra (2) lecture, we had $\sigma$-linear-forms with $(\sigma(z))^2 = 1$ and usually $\sigma(z) = \bar{z}$ or $\sigma(z) = 1$, depending on the field. But that took me a while to understand, so the didactic purpose is questionable ;-) $$$$ P.S.: What fields have inner products which are indefinite? Never heard about that, actually I reckon it's an axiom... $\endgroup$ – AlexR Aug 26 '13 at 19:33
  • $\begingroup$ @AlexR Well, it all depends on how free you are with the terms. If you take positive definiteness as an axiom, then of course it doesn't apply. I'll try to clear this up in my solution. $\endgroup$ – rschwieb Aug 26 '13 at 19:45
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A dot product is a very specific inner product that works on $\Bbb{R}^n$ (or more generally $\Bbb{F}^n$, where $\Bbb{F}$ is a field) and refers to the inner product given by

$$(v_1, ..., v_n) \cdot (u_1, ..., u_n) = v_1 u_1 + ... + v_n u_n$$

More generally, an inner product is a function that takes in two vectors and gives a complex number, subject to some conditions.

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In my experience, inner product is defined on vector spaces over a field $\mathbb K$ (finite or infinite dimensional). Dot product refers specifically to the product of vectors in $\mathbb R^n$, however.

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  • $\begingroup$ I see.... Thanks! $\endgroup$ – CuriousGeorge119 Aug 26 '13 at 19:51
  • $\begingroup$ you are welcome! $\endgroup$ – Avitus Aug 26 '13 at 19:53
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Let...

  • $c\in\mathbb{C}$
  • $\vec{u}$, $\vec{v}\in\mathbb{R}_{m\times1}$
  • $A$, $B\in\mathbb{C_{m\times n}}$
  • Complex conjugate of $c = \overline{c} = c^*$
  • Absolute value of $c = \left|c\right| = \sqrt{\overline{c}c}$

Now, according to wiki...

  • Dot product of $\vec{u}$ & $\vec{v} = \vec{u}\cdot\vec{v} = \vec{u}^T\vec{v}$.
  • Inner product of $\vec{u}$ & $\vec{v} = \left<\vec{u},\vec{v}\right> = \vec{u}^T\vec{v}$.

As mentioned in the chosen answer, the inner product is sometimes used as a generalization of the dot product (& the referenced article talks about this) but, in the interest of brevity/utility, I advise you to consider them equivalent unless you have reason to believe otherwise (i.e., strange notation, explicit definitions, context, etc.).

As silly as this sounds, I tend to use $\left<\vec{u},\vec{v}\right>$ in place of $\vec{u}\cdot\vec{v}$ simply bc it is more explicit/identifiable (i.e., my brain is hardwired to associate "$\cdot$" w/ scalar multiplication) &, if my memory serves me, I picked up the habit from Dr. Strang.

P.S. To extend the ideas of dot/inner products to arbitrary real/complex matrices, checkout the Frobenius inner product...

$$ A:B = \left<A,B\right>_F = \text{tr}\left(\overline{A}^TB\right) $$

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