Linear Independence

Linear Independence

In linear algebra, two slightly different notions of linear independence used: the linear independence of a family of vectors, and the linear independence of a set of vectors.

  • A family of vectors is linearly independent family if none of them can be written as a linear combination of finitely many other vectors in the family. A family of vectors which is not linearly independent is called linearly dependent.
  • A set of vectors is linearly independent set if the set (regarded as a family indexed by itself) is a linearly independent family.

These two notions aren't equivalent: the difference being that in a family we allow repeated elements, while in a set we don't. For example if is a vector space, then the family such that and is a linearly dependent family, but the singleton set of the images of that family is wich is a linearly independent set.

Both notions are important and used in common, and sometimes even confused in the literature.

For instance, in the three-dimensional real vector space we have the following example.

\begin{matrix} \mbox{independent}\qquad\\ \underbrace{ \overbrace{ \begin{bmatrix}0\\0\\1\end{bmatrix}, \begin{bmatrix}0\\2\\-2\end{bmatrix}, \begin{bmatrix}1\\-2\\1\end{bmatrix} }, \begin{bmatrix}4\\2\\3\end{bmatrix} }\\ \mbox{dependent}\\ \end{matrix}

Here the first three vectors are linearly independent; but the fourth vector equals 9 times the first plus 5 times the second plus 4 times the third, so the four vectors together are linearly dependent. Linear dependence is a property of the family, not of any particular vector; for example in this case we could just as well write the first vector as a linear combination of the last three.

In probability theory and statistics there is an unrelated measure of linear dependence between random variables.

Read more about Linear Independence:  Definition, Geometric Meaning, Example II, Example III, Example IV, Projective Space of Linear Dependences, Linear Dependence Between Random Variables

Famous quotes containing the word independence:

    “Traditionally in American society, men have been trained for both competition and teamwork through sports, while women have been reared to merge their welfare with that of the family, with fewer opportunities for either independence or other team identifications, and fewer challenges to direct competition. In effect, women have been circumscribed within that unit where the benefit of one is most easily believed to be the benefit of all.”
    —Mary Catherine Bateson (b. 1939)