Determine if Linearly Dependent

Problem

S={[[2],[−1],[3],[4],[2]],[[6],[2],[−1],[3],[4]]}

Solution

1. Identify the number of vectors in the set. The set S contains two vectors, (v_1) and (v_2)

2. Recall the condition for linear dependence of two vectors. A set of two vectors is linearly dependent if and only if one vector is a scalar multiple of the other.

3. Set up the scalar equation to check for a constant k such that (v_2)=k*(v_1)

[[6],[2],[−1],[3],[4]]=k*[[2],[−1],[3],[4],[2]]

4. Solve for k using the first component.

6=2*k

k=3

5. Verify the scalar k with the second component.

2=−1*k

2=−1*(3)

2≠−3

6. Conclude that since there is no single scalar k that satisfies the equation for all components, the vectors are not multiples of each other. Therefore, the set is linearly independent.

Final Answer

S=Linearly Independent

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