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

columns of a matrix are linearly independent | 0.16 | 0.2 | 3337 | 70 | 44 |

columns | 1.97 | 0.5 | 2591 | 83 | 7 |

of | 0.63 | 0.3 | 1340 | 7 | 2 |

a | 1.58 | 0.1 | 6346 | 99 | 1 |

matrix | 1.69 | 0.3 | 3678 | 71 | 6 |

are | 0.5 | 0.4 | 3191 | 19 | 3 |

linearly | 1.17 | 0.2 | 2350 | 78 | 8 |

independent | 1.41 | 0.4 | 211 | 79 | 11 |

When I say linear independent I mean not linearly dependent with any other column or any combination of other columns in the matrix. For example: In this matrix we know that column 1 is linear independent and columns 2 and 3 are dependent.

A wide matrix (a matrix with more columns than rows) has linearly dependent columns. For example, four vectors in R3are automatically linearly dependent. Note that a tall matrix may or may not have linearly independent columns. Facts about linear independence

For a 3x3 matrix, such as A To find if rows of matrix are linearly independent, we have to check if none of the row vectors (rows represented as individual vectors) is linear combination of other row vectors. Row vectors of matrix A Turns out vector a3is a linear combination of vector a1and a2. So, matrix Ais not linearly independent.

By knowing the set of three vectors is linearly independent, we know that the third column vector cannot be written as a linear combination of the first column vector and the second column vector. That is, there do not exist c 1, c 2 ∈ R such that v 3 = c 1 v 1 + c 2 v 2. ( v i is my notation for the i -th column vector.)