The Inner Product on a Vector Space

An inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. The dot product is a vector space, as is the magnitude of the cross product of two vectors. If  
\[\mathbf{v}_1, \: \mathbf{v}_2\]
  are vectors in a vector space  
  is a linear transformation sending elements of the space onto other elements, with associated matrix  
, then  
\[\mathbf{v}^T_1 M \mathbf{v}_2 \]
  is the inner product of  
  may be any matrix, including the zero matrix.
The inner product is symmetric:  
\[\langle \mathbf{v}_1, \mathbf{v}_2 \rangle =\langle \mathbf{v}_2, \mathbf{v}_1 \rangle\]

The inner product is positive definite:  
\[\langle \mathbf{v}, \mathbf{v} \rangle \geq 0\]
\[\langle \mathbf{v}, \mathbf{v} \rangle = 0 \leftrightarrow \mathbf{v}= \mathbf{0}\]

The inner product is linear in both arguments:  
\[\langle \alpha \mathbf{v}_1, \mathbf{v}_2 \rangle =\langle \mathbf{v}_1, \alpha \mathbf{v}_2 \rangle = \alpha \langle \mathbf{v}_1, \mathbf{v}_2 \rangle\]

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