The effect of this rotation on a 2D vector in the plane can be determined by multiplying the vector by this transformation matrix,

(9) |

More generally, the product of a matrix and a column vector may be
written as

(10) |

(11) | |||

Notice that each element of the product, , is just the dot product of the th row of the matrix with the vector. It is also possible to think of the column vector as being a linear combination of the columns of the matrix , with each column having a weight .

It is also possible to ``left-multiply'' a matrix by a row vector, like
this:

(12) |

(13) | |||

This time, each element of the product, , is the dot product of the vector on the left with the th column of the matrix. The row vector may be thought of as a linear combination of the rows of the matrix , with each row having a weight of .

From the information given, it should be obvious that a row vector times a matrix times a column vector yields a number (scalar).

Now consider the multiplication of two matrices and ,
which yields a new matrix, :

(14) |

where

(15) |

The *transpose* of a matrix , typically denoted
, is obtained simply by swapping elements across the diagonal,
. For example,

(16) |

For complex numbers, usually the *complex conjugate transpose*, or
*adjoint*, is more useful. The adjoint of a matrix is often
denoted
. For example,

(17) |

(18) |