**1.**

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number.

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In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number.

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Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.

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Dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar a,.

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Dot product fulfills the following properties if a, b, and c are real vectors and r is a scalar.

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The dot product is a scalar in this sense, given by the formula, independent of the coordinate system.

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When vectors are represented by column vectors, the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H:.

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The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector.

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However, the complex dot product is sesquilinear rather than bilinear, as it is conjugate linear and not linear in a The dot product is not symmetric, since.

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Complex dot product leads to the notions of Hermitian forms and general inner product spaces, which are widely used in mathematics and physics.

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An inner Dot product space is a normed vector space, and the inner Dot product of a vector with itself is real and positive-definite.

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Dot product is defined for vectors that have a finite number of entries.

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Inner Dot product between a tensor of order n and a tensor of order m is a tensor of order, see Tensor contraction for details.

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