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# s08hw5 - Week 5 2.2 Subspaces/Spanning(continued 2.3...

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Week 5: 2.2 Subspaces/Spanning (continued) 2.3 Independence/Bases

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2 Linear independence/Bases and Dimension Vectors ±v 1 ,±v 2 , . . . ,±v k in a vector space V are linearly dependent if there are scalars c 1 , c 2 , . . . , c k so ± 0 = c 1 ±v 1 + c 2 ±v 2 + ··· + c k ±v k , with at least one c j ± = 0 . Such a vector equation is said to be a relation of linear dependence among the ±v 1 ,±v 2 , . . . ,±v k . A collection of vectors for which the only solution of the vector equation is the trivial solution, c 1 = 0 , . . . , c k = 0 is said to be linearly independent. Examples: (1) ± i, ± j in R 2 ; and (2) { ± i, ± j, ± k } in R 3 ; are linearly independent. Veriﬁcation: c 1 ± i + c 2 ± j + c 3 ± k = c 1 (1 , 0 , 0) + c 2 (0 , 1 , 0) + c 3 (0 , 0 , 1) = ( c 1 , c 2 , c 3 ) = (0 , 0 , 0) = ± 0 , only when c 1 = 0 , c 2 = 0 , c 3 = 0 .
Our main objective is the deﬁnition of basis. Vectors ±v 1 ,±v 2 , . . . ,±v k in a vector space V that are are both (1) a spanning set for V and (2) linearly independent are called a

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s08hw5 - Week 5 2.2 Subspaces/Spanning(continued 2.3...

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