In linear algebra, a set of vectors in a vector space over a field are linearly independent if there do not exist scalars not all equal to zero such that Otherwise, the vectors are said to be linearly dependent.
A basis of a vector space is a maximal set of linearly independent vectors, that is, if are a basis, then for any vector are linearly dependent.
Any eigenvectors corresponding to different eigenvalues (with respect to a linear map ) are linearly independent. This can be proved by induction. Suppose are eigenvectors corresponding to distinct eigenvalues , and that there exists a statement of linear dependency Multiplying both sides by and applying to both sides, respectively, yields Subtracting the two equations yields which is a statement of linear dependency among .
- Show that the Hilbert matrix has a non-zero determinant.
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