Let V be a vector space over a field F, [tex]v_{1}, \cdots, v_{n} \in V[/tex] and [tex]\alpha_{1}, \cdots, \alpha_{n} \in F[/tex]. Further on, let the set [tex]\left\{v_{1}, \cdots, v_{n}\right\}[/tex] be linearly independent, and b be a vector defined with [tex]b=\sum_{i=1}^n \alpha_{i}v_{i}[/tex]. One has to find necessary and sufficient conditions on the scalars [tex]\alpha_{1}, \cdots, \alpha_{n}[/tex] such that the set S=[tex]\left\{b, v_{2}, \cdots, v_{n}\right\}[/tex] is linearly independent, too.(adsbygoogle = window.adsbygoogle || []).push({});

Well, I just used a simple proposition which states that a set is dependent if there exists at least one vector from that set which can be shown as a linear combination of the rest of the vectors from the same set. So, obviously, for [tex]\alpha_{1} = 0[/tex], the set S is dependent, which makes [tex]\alpha_{1} \neq 0[/tex] a necessary condition for S to be independent. Further on, a sufficient condition would be [tex]\alpha_{1} = \cdots = \alpha_{n} = 0[/tex], which leaves us with the set S\{b}. This set must be linearly independent, since it is a subset of {v1, ..., vn}, which we know is linearly independent.

I may be boring, but I'm just checking if my reasoning is allright..

Edit. I just realized, for [tex]\alpha_{1} = \cdots = \alpha_{n} = 0[/tex] we have b = 0, which makes the set dependent! So [tex]\alpha_{1} \neq 0[/tex] is a necessary condition, but what's the sufficient condition? Is it that at least one of the scalars [tex]\alpha_{2}, \cdots, \alpha_{n}[/tex]must not be equal zero?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Linear independence again

Loading...

Similar Threads for Linear independence again | Date |
---|---|

I Measures of Linear Independence? | Dec 14, 2017 |

I Proving a set is linearly independant | Apr 14, 2017 |

I Question On Linear Independence | Nov 2, 2016 |

I Expanding linear independent vectors | Sep 11, 2016 |

I Vector components, scalars & coordinate independence | Jul 26, 2016 |

**Physics Forums - The Fusion of Science and Community**