Linear map: Difference between revisions

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: <math>f(\mathbf{v}_j)=a_{1j} \mathbf{w}_1 + \cdots + a_{mj} \mathbf{w}_m.</math>

Thus, the function ''f'' is entirely determined by the values of ''a_ij''. If we put these values into an ''m'' × ''n'' matrix ''M'', then we can conveniently use it to compute the value of ''f'' for any vector in ''V''. To get M, every column j of M is a vector whose coordinates are

: <math>(a_{1j},...,a_{mj})^T</math>

If we put these values into an ''m'' × ''n'' matrix ''M'', then we can conveniently use it to compute the value of ''f'' for any vector in ''V''. For if we place the values of ''c''₁, ..., ''c_n'' in an ''n'' × 1 matrix ''C'', we have ''MC'' = the ''m'' × 1 matrix whose ''i''th element is the coordinate of ''f''('''v''') which belongs to the base '''w'''_''i''.

corresponding to f('''v'''_j) as defined above. To define it more clearly, for some column j,

A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.

: <math>\mathbf{M}=\begin{pmatrix} & & & a_{1j} & &\\ & & &.& &\\ & *& &.& & *\\ & & &.& & \\ & & &a_{mj}& &\end{pmatrix}</math>

The symbol * denotes that there are other columns which together with column j make up a total of n columns of M. A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.

== Examples of linear transformation matrices ==