Question

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 independent
2. S₁ is linearly independent but S₂ is linearly dependent
3. S₁ is linearly dependent but S₂ is linearly independent
4. Both S₁ and S₂ are linearly dependent

Answer 1

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.