Linearly independent

In linear algebra, a set of vectors $v_1, v_2, \ldots, v_n$ in a vector space $V$ over a field $K$ are linearly independent if there do not exist scalars $c_1, c_2, \ldots, c_n \in K$ not all equal to zero such that $c_1v_1 + c_2v_2 + \cdots + c_nv_n = O.$ Otherwise, the vectors are said to be linearly dependent.

In $\mathbb{R}^m$ , vectors $\bold{v}_1, \ldots, \bold{v}_n$ are linearly independent iff their determinant $D(\bold{v}_1, \ldots, \bold{v}_n) = 0$ .

Examples

A basis of a vector space $V$ is a maximal set of linearly independent vectors, that is, if $\{v_1, \ldots, v_n\}$ are a basis, then $\{v_1, \ldots, v_n, w\}$ for any vector $w \in V$ are linearly dependent.

Any eigenvectors corresponding to different eigenvalues (with respect to a linear map $L$ ) are linearly independent. This can be proved by induction. Suppose $v_1, v_2, \ldots, v_n$ are eigenvectors corresponding to distinct eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ , and that there exists a statement of linear dependency $\sum c_iv_i = O.$ Multiplying both sides by $\lambda_1$ and applying $L$ to both sides, respectively, yields $\sum \lambda_1c_iv_i = O,\qquad L\left(\sum c_iv_i\right) = \sum \lambda_ic_iv_i = O.$ Subtracting the two equations yields $\sum (\lambda_1-\lambda_i)c_iv_i,$ which is a statement of linear dependency among $v_2, \ldots, v_n$ .

Problems

Show that the Hilbert matrix has a non-zero determinant.

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Linearly independent

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