MATH-M 303 Lecture Notes - Lecture 6: Gaussian Elimination, Linear Combination, Free Variables And Bound Variables
Document Summary
In space/plane, vectors truly point in different directions. If no free variables present, vectors linearly independent: no vector in set is linear combination of the others, property that non-empty set of vectors may or may not have. Linear independence- set of vectors (cid:2778),(cid:2779), , linearly independent if vector equation (cid:2869)(cid:2779)+(cid:2870)(cid:2779)+(cid:1710)+=(cid:2777) has only trivial solution =(cid:2777), ie. (cid:2869),(cid:2870), ,=(cid:882) If =[(cid:2778) (cid:2779) ], then =(cid:2777) has same solution as (cid:2869)(cid:2869)+(cid:2870)(cid:2870)+(cid:1710)+=(cid:2777) Columns of linearly independent iff =(cid:2777) has only trivial solution and iff no free variables. Linear dependence- vectors (cid:2778),(cid:2779), , linearly dependent if (cid:2869)(cid:2779)+(cid:2870)(cid:2779)+(cid:1710)+=(cid:2777) has non- trivial solutions, ie. if (cid:2869),(cid:2870), , not all zero: (cid:2869)(cid:2779)+(cid:2870)(cid:2779)+(cid:1710)+=(cid:2777) where some scalar(s) not zero known as dependence relation, form matrix =[(cid:2778) (cid:2779) ], reduce to ef. If dependent, any nontrivial solution to =(cid:2777) gives dependence relation: can also interpret homogeneous =(cid:2777) as vector equation (cid:2869)(cid:2778)+(cid:2870)(cid:2779)+(cid:1710)+=(cid:2777) (cid:2869)(cid:4666)(cid:883),(cid:884),(cid:885)(cid:4667)+(cid:2870)(cid:4666)(cid:886),(cid:887),(cid:888)(cid:4667)+(cid:2871)(cid:4666)(cid:884),(cid:883),(cid:882)(cid:4667)=(cid:4666)(cid:882),(cid:882),(cid:882)(cid:4667, always have trivial solution =(cid:4666)(cid:882),(cid:882),(cid:882)(cid:4667); linear independence means this is the only solution.
