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1.5 Linear Dependence

# 1.5 Linear Dependence - 1.5 1.5 Linear Dependence and...

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1.5. 1.5. Linear Dependence and Linear Independence 1 . (a) If S is a linearly dependent set, then each vector in S is a linear combina- tion of other vector in S . Ans : F (Example) V = R 3 , S = { e 1 = (1 , 0 , 0) , e 2 = (0 , 1 , 0) , e 3 = (0 , 0 , 1) , e 4 = (1 , 1 , 0) } { e 1 , e 2 , e 3 } : linearly independent { e 1 , e 2 , e 4 } : linearly dependent (b) Any set containing the zero vector is linearly dependent. Ans : T ( ) a F, 0 = a · 0 , a 6 = 0 (c) The empty set is linearly dependent. Ans : F ( ) linearly dependent set must be non-empty. (d) Subsets of linearly dependent sets are linearly dependent. Ans : F ( ) theorem 1.6 (e) Subsets of linearly independent sets are linearly independent. Ans : T ( ) the corollarly from theorem 1.6 (f) If a 1 x 1 + a 2 x 2 + · · · + a n x n = 0 and x 1 , x 2 , · · · , x n are linearly independent, then all the scalars a i are zero. Ans : T ( ) from the definition. 2 . (a), (d), (e), (g), (h), (j) : linearly independent (b), (c), (f), (i) : linearly dependent 24 PNU-MATH

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1.5. 3 . In M 2 × 3 ( F ), prove that the set 1 1 0 0 0 0 , 0 0 1 1 0 0 , 0 0 0 0 1 1 , 1 0 1 0 1 0 , 0 1 0 1 0 1 is linearly dependent. ( ) a + d = 0 , b + d = 0 , c + d = 0 , a + e = 0 , b + e = 0 , c + e = 0 a = - d, b = - d, c = - d, d, e = d the given set is linearly dependent. 4. In F n , let e j denote the vector whose j th coordinate is 1 and whose other coordinates are 0. Prove that { e 1 , e 2 , · · · , e n } is linearly independent. ( ) ( a 1 , a 2 , · · · , a n ) = a 1 e 1 + a 2 e 2 + · · · + a n e n = (0 , · · · , 0) a 1 = a 2 = · · · = a n = 0 { e 1 , e 2 , · · · , e n } is linearly independent.
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1.5 Linear Dependence - 1.5 1.5 Linear Dependence and...

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