Definition:Dot Product/General Context

From ProofWiki
Jump to navigation Jump to search

Definition

Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $n$ dimensions:

\(\ds \mathbf a\) \(=\) \(\ds \sum_{k \mathop = 1}^n a_k \mathbf e_k\)
\(\ds \mathbf b\) \(=\) \(\ds \sum_{k \mathop = 1}^n b_k \mathbf e_k\)


where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.


The dot product of $\mathbf a$ and $\mathbf b$ is defined as:

\(\ds \mathbf a \cdot \mathbf b\) \(:=\) \(\ds \sum_{k \mathop = 1}^n a_k b_k\)
\(\ds \) \(=\) \(\ds a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i \mathop = 1}^n a_i b_i\)


If the vectors are represented as column matrices:

$\mathbf a = \begin {bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end {bmatrix} , \mathbf b = \begin {bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end {bmatrix}$

we can express the dot product as:

$\mathbf a \cdot \mathbf b = \mathbf a^\intercal \mathbf b$

where:

$\mathbf a^\intercal = \begin {bmatrix} a_1 & a_2 & \cdots & a_n \end {bmatrix}$ is the transpose of $\mathbf a$
the operation between the matrices is the matrix product.


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Dot_Product/General_Context&oldid=726896"