Keyword | CPC | PCC | Volume | Score | Length of keyword |
---|---|---|---|---|---|

linearly independent calculator for vectors | 1.91 | 0.1 | 3161 | 38 | 43 |

linearly | 0.63 | 0.6 | 3872 | 8 | 8 |

independent | 1.37 | 0.6 | 1344 | 19 | 11 |

calculator | 0.32 | 0.1 | 139 | 8 | 10 |

for | 1.05 | 0.3 | 5142 | 95 | 3 |

vectors | 1.06 | 0.1 | 7865 | 95 | 7 |

set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero The set is of course dependent if the determinant is zero. Example The vectors <1,2> and <-5,3> are linearly independent since the matrix has a non-zero determinant. Example

Note that because a single vector trivially forms by itself a set of linearly independent vectors. Moreover, because otherwise would be linearly independent, a contradiction. Now, can be written as a linear combination of : where are scalars and they are not all zero (otherwise would be zero and hence not an eigenvector).

Two non-collinear non-zero vectors are always linearly independent . Three coplanar vectors are always linearly dependent. Three non-coplanar non-zero vectors are always linearly independent. More than 3 vectors are always linearly dependent.