Editing Row and column vectors (section)

== Operations ==

[[Matrix multiplication]] involves the action of multiplying each row vector of one matrix by each column vector of another matrix. 

The [[dot product]] of two column vectors {{math|'''a''', '''b'''}}, considered as elements of a coordinate space, is equal to the matrix product of the transpose of {{math|'''a'''}} with {{math|'''b'''}},

<math display="block">\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^\intercal \mathbf{b} = \begin{bmatrix}
    a_1  & \cdots  & a_n
\end{bmatrix} \begin{bmatrix} 
    b_1 \\ \vdots \\ b_n
\end{bmatrix} = a_1 b_1 + \cdots + a_n b_n \,, </math>

By the symmetry of the dot product, the [[dot product]] of two column vectors {{math|'''a''', '''b'''}} is also equal to the matrix product of the transpose of {{math|'''b'''}} with {{math|'''a'''}},

<math display="block">\mathbf{b} \cdot \mathbf{a} = \mathbf{b}^\intercal \mathbf{a} = \begin{bmatrix}
    b_1  & \cdots  & b_n
\end{bmatrix}\begin{bmatrix} 
    a_1 \\ \vdots \\ a_n
\end{bmatrix} = a_1 b_1 + \cdots + a_n b_n\,. </math>

The matrix product of a column and a row vector gives the [[outer product]] of two vectors {{math|'''a''', '''b'''}}, an example of the more general [[tensor product]]. The matrix product of the column vector representation of {{math|'''a'''}} and the row vector representation of {{math|'''b'''}} gives the components of their dyadic product,

<math display="block">\mathbf{a} \otimes \mathbf{b} = \mathbf{a} \mathbf{b}^\intercal = \begin{bmatrix}
    a_1 \\ a_2 \\ a_3
\end{bmatrix}\begin{bmatrix} 
    b_1 & b_2 & b_3
\end{bmatrix} = \begin{bmatrix} 
a_1 b_1 & a_1 b_2 & a_1 b_3 \\
a_2 b_1 & a_2 b_2 & a_2 b_3 \\
a_3 b_1 & a_3 b_2 & a_3 b_3 \\
\end{bmatrix} \,, </math>

which is the [[transpose]] of the matrix product of the column vector representation of {{math|'''b'''}} and the row vector representation of {{math|'''a'''}},

<math display="block">\mathbf{b} \otimes \mathbf{a} = \mathbf{b} \mathbf{a}^\intercal = \begin{bmatrix}
    b_1 \\ b_2 \\ b_3
\end{bmatrix}\begin{bmatrix} 
    a_1 & a_2 & a_3
\end{bmatrix} = \begin{bmatrix} 
b_1 a_1 & b_1 a_2 & b_1 a_3 \\
b_2 a_1 & b_2 a_2 & b_2 a_3 \\
b_3 a_1 & b_3 a_2 & b_3 a_3 \\
\end{bmatrix} \,. </math>