Mathematics-Online lexicon: Matrix of a Linear Map

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	Mathematics-Online lexicon:
	Matrix of a Linear Map

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

A linear map $\alpha: V \longmapsto W$ between two

-vector spaces with bases $E = \{e_1,\dots,e_n\}$ and $F = \{f_1,\dots,f_m\}$ is uniquely determined by the images of the basis vectors

$\displaystyle \alpha(e_j) =: a_{1,j} f_1 + \dots + a_{m,j} f_m\, .$

We obtain the matrix representation

$\displaystyle w_F = Av_E \longleftrightarrow w_i = \sum_{i=1}^n a_{i,j} v_j,\quad i=1,\ldots,m\, ,$

where

and

denote the coordinates relative to the bases

and

, resp.

see also:

automatically generated 3/30/2005