Lecture 6 Notes

A set containing the 0 vector theorem 9 a set of

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Unformatted text preview: zero vector is linearly dependent. Proof: Renumber the vectors so that v 1  ____. Then ____v 1  _____v 2    _____v p  0 which shows that S is linearly ________________. 4. A Set Containing Too Many Vectors Theorem 8 If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. I.e. any set v 1 , v 2 , , v p in R n is linearly dependent if p  n. Outline of Proof: A v1 v2  vp is n p Suppose p  n. Í Ax  0 has more variables than equations Í Ax  0 has nontrivial solutions Ícolumns of A are linearly dependent 9 EXAMPLE With the least amount of work possible, decide which of the following sets of vectors are linearly independent and give a reason for each answer. 3 a. 2 9 , 1 6 4 12345 b. Columns of 67890 98765 43218 10 3 c. 2 1 9 , 6 3 0 , 0 0 8 d. 2 1 4 11 Characterization of Linearly Dependent Sets EXAMPLE Consider the set of vectors v 1 , v 2 , v 3 , v 4 in R 3 in the following diagram. Is the set linearly dependent? Explain v3 x3 v2 x2 x1 v1 v4 12 Theorem 7 An indexed set S  v 1 , v 2 , , v p of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others. In fact, if S is linearly dependent, and v 1 0, then some vector v j (j 2) is a linear combination of the preceding vectors v 1 , , v j 1 . 13...
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