Inner product and dot product

  • Thread starter Taturana
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  • #1
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Could someone explain me the difference between the inner product and the dot product?

Thanks all
 

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  • #2
mathman
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There really isn't any. The terms are used more or less interchangeably. For 2 or 3 dimensions dot product is usually used, while for higher dimensions and Hilbert space, inner product is usually used, but they mean the same thing.
 
  • #3
HallsofIvy
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There is, in a sense, a technical difference. The "dot product" is defined in any [itex]R^n[/itex] (or [itex]C^n[/itex] while an "inner product" is any function from a pair of vectors, in any vector space, to the real numbers, satisfying
1) <u, v>= <v, u> (or, if the vector space is over the complex numbers, the complex conjugate of <v, u>.
2) <u+ v, w>= <u, w>+ <v, w>.
3) <ku, v>= k<u, v> for any number k.

Of course, any dot product is an inner product.

Going the other way, if we choose an orthonormal basis for the vector space, [itex]\{v_1, v_2, v_3, \cdot\cdot\cdot, v_n\}[/itex] any vectors u, and v, can be written as a linear combination of them, and then write the inner product as a dot product of those coefficients. The theoretical "meat" of the Gram-Schmidt procudure (that such an orthonormal basis exists) is that every inner product is a dot product- in some basis.
 

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