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Consider two subsets of R3 given as S1 = {[7, 7, 7]} and S2 = {[ 0, 0, 0]}. Which of the following statements is true?1. Both S2 and S2 are linearly independent2. S1 is linearly independent but S2 is linearly dependent3. S1 is linearly dependent but S2 is linearly independent4. Both S1 and S2 are linearly dependent |
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Answer» Correct Answer - Option 2 : S1 is linearly independent but S2 is linearly dependent Concept: A set of vectors {v1, v2,…, vp} in a vector space V is said to be linearly independent if the vector equation c1v1 + c2v2 +…+ cpvp = 0 has only one trivial solution c1 = 0, c2 = 0,…, cp = 0; The set is said to be linearly dependent if there exists weights c1, c2,…, cp not all 0, such that c1v1 + c2v2 +…+ cpvp = 0 Calculation: Given S1 = {(7,7,7)} ⇒ v1 = (7,7,7) S2 = (0,0,0) A set containing zero vector is linearly dependent ⇒ S2 is linearly dependent Let S1 = (v1) and the vector equation be av1 = 0 ⇒ 7a = 0 ⇒ a = 0 Since, the constant is zero, S1 will be linearly independent. |
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