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# Linear transformations and matrices.

We suppose now that U and V are finite dimensional, and that , . Let be a libear mapping.

Choose bases: for U and for V.

Take a vector in U; there exists a unique n-tuple of scalars such that . Then:

Therefore the linear mapping is totally determined by the data of the vectors .

Denote: .

We represent this data by the matrix:

If the coordinates of w.r.t. the given basis of U are and if the coordinates of w.r.t. the given basis of V are , they are related by the formulas:

We denote this in the following way:

Example 6.3.1   the vector space of polynomial in one real indeterminate x, with coefficients in and such that .

A basis for U: p1(x)=1, [2(x)=x, p3(x)=x2, p4(x)=x3.

.

We have: , , , . Thus, the matrix of w.r.t the given basis is: .

Next: The algebra of matrices. Up: Linear mappings. Previous: Kernel and Image of
Noah Dana-Picard
2001-02-26