» Required Reading » Table of Contents » Chapter 4 4.2 Matrices and Transformations on Vectors; the Meaning of 0 Determinant
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Matrices and determinants appear in two other important contexts; one is in solving simultaneous linear equations in several variables. The other is in representing linear transformations of vectors.

In the latter context a matrix represents the transformation that takes the column basis vectors into the vectors that are the corresponding columns of the matrix.

Sums of original basis vectors are transformed into the same sums of the corresponding columns.

When the determinant of a matrix is zero, the volume of the region with sides given by its columns or rows is zero, which means the matrix considered as a transformation takes the basis vectors into vectors that are linearly dependent and define 0 volume.

This happens, the determinant is zero, when the columns (and rows) of the matrix are linearly dependent.