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A (finite) set of vectors <math>v_1, v_2...v_m</math>is said to be linearly independent if and only if the equality <math>k_1v_1+k_2v_2+...k_mv_m=0</math> is true [[exactly when]] all the k values are 0. | A (finite) set of vectors <math>v_1, v_2...v_m</math>is said to be linearly independent if and only if the equality <math>k_1v_1+k_2v_2+...k_mv_m=0</math> is true [[exactly when]] all the k values are 0. | ||
| − | This is equivalent to saying you can't come up with any [[linear combination]] of <math>v_1</math> and <math>v_2</math> that equals v_3, or <math>v_1...v_3</math> that equals <math>v_4</math>... or <math>v_1...v_{m-1}</math> that equals <math>v_m</math>. | + | This is equivalent to saying you can't come up with any [[linear combination]] of <math>v_1</math> and <math>v_2</math> that equals <math>v_3</math>, or <math>v_1...v_3</math> that equals <math>v_4</math>... or <math>v_1...v_{m-1}</math> that equals <math>v_m</math>. |
If a set of vectors are not linearly independent, then they are linearly dependent. | If a set of vectors are not linearly independent, then they are linearly dependent. | ||
Latest revision as of 16:28, 11 March 2013
When are vectors linearly independent?
A (finite) set of vectors $ v_1, v_2...v_m $is said to be linearly independent if and only if the equality $ k_1v_1+k_2v_2+...k_mv_m=0 $ is true exactly when all the k values are 0.
This is equivalent to saying you can't come up with any linear combination of $ v_1 $ and $ v_2 $ that equals $ v_3 $, or $ v_1...v_3 $ that equals $ v_4 $... or $ v_1...v_{m-1} $ that equals $ v_m $.
If a set of vectors are not linearly independent, then they are linearly dependent.
