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Mathematics-Online course: Linear Algebra - Matrices - Linear Maps

The Matrix of a Linear Map

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A linear map $ \alpha: V \longmapsto W$ between two $ K$-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 $ v_j$ and $ w_i$ denote the coordinates relative to the bases $ E$ and $ F$, resp.

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  automatically generated 4/21/2005