MATH2107-4-31 - Section 4.3 Linearly independent sets bases...

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Section 4.3 Linearly independent sets; bases Linear Algebra and its Applications David C. Lay — Winter 2008 – p. 1/14
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Let’s start with a vector space V and an ordered subset { v 1 , · · · , v p } ⊂ V . The ideas are similar to those for R n . { v 1 , · · · , v p } ⊂ V is said to be linearly independent if c 1 v 1 + · · · + c p v p = 0 (1) has only the trivial solution c 1 = · · · = c p . The set { v 1 , · · · , v p } is said to be linearly dependent if (1) has a non-trivial solution. i.e. c i negationslash = 0 for at least one i . When { v 1 , · · · , v p } is linearly dependent there is a linear dependence relation among the vectors v 1 ,..., v p . — Winter 2008 – p. 2/14
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Theorem { v 1 , · · · , v p } with v 1 negationslash = 0 is linearly dependent if and only if some v j ,j > 1 is a linear combination of the preceeding vectors, v 1 ,..., v j 1 . — Winter 2008 – p. 3/14
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Example e 1 = 1 0 0 , e 2 = 0 1 0 , e 3 = 0 0 1 . 1. Show { e 1 , e 2 , e 3 } is linearly independent.
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