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NENE 112112 LinLinear ear alalgebra gebra for for nananonotechtechnnololoogy gy engengineeineeringring 6.5 Linear dependence and independence of a collection of vectors DoDouuglas glas WilheWilhelm lm HaHarrdederr, , LELLEL, , M.MM.Mathath.. dwhdwhaarder@rder@uwauwateterloorloo.ca.ca dwdwhhaarrder@der@gmailgmail.c.coomm Linear dependence and independence of vectors • In this topic, we will Introduction – Define linear dependence and independence of a given vector v on a collection of vectors – Determine how to check if a vector linearly depends on others – Define the rank of a matrix – Define a linearly dependent collection of vectors, and a linearly independent collection of vectors – Determine how to check if a collection of vectors is linearly dependent or independent – Consider a few results 2 Linear dependence and independence of vectors Definition • A vector u is said to be linearly dependent on a collection of nvectors v1, …, vn if u can be written as a linear combination of the vectors v , …, v 1 n – That is, u is linearly dependent on v1, …, vn if the problem v + + v =u 11 nn has at least one solution • If u is not linearly dependent on a collection of nvectors v1, …, vn, we say u is linearly independent of v1, …, vn – That is, the problem v + + v =u has no solutions 11 nn 3 Linear dependence and independence of vectors The zero vector • Note that the vector 0m is linearly dependent on any collection of nm-dimensionalvectors as v + + v =0 11 n n m always has at least one solution: = = =0 1 n 4
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