Exercise_5

# Exercise_5 - -a b c 6 = 0 2.No If there are c 1,c 2,c 3 ∈...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Exercise 5 (In Class) Name: Student ID: Class: Let V be the vector space R 3 and let v 1 = 1 1 0 , v 2 = 1 0 1 , v 3 = 0 - 1 1 1. Do v 1 , v 2 , v 3 span V ? Explain. 2. Are the vectors v 1 , v 2 , v 3 linearly independent? Explain. Solution: 1.No. For any v = a b c , if there are c 1 ,c 2 ,c 3 R , such that c 1 v 1 + c 2 v 2 + c 3 v 3 = v . that is , c 1 + c 2 = a c 1 - c 3 = b c 2 + c 3 = c , whose reduced row echelon form is 1 0 - 1 b 0 1 1 a - b 0 0 0 - a + b + c , which is not consistent if
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Unformatted text preview: -a + b + c 6 = 0 . 2.No. If there are c 1 ,c 2 ,c 3 ∈ R , such that c 1 v 1 + c 2 v 2 + c 3 v 3 = . that is ,    c 1 + c 2 = 0 c 1-c 3 = 0 c 2 + c 3 = 0 , whose reduced row echelon form is   1 0-1 0 1 1 0 0   , which has inﬁnitely many solutions { r (1 ,-1 , 1) : r ∈ R } . Thus, c 1 , c 2 and c 3 are not all zero....
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