Determine if Linearly Dependent

Problem

S={[[2],[−1],[3],[1],[2]],[[1],[2],[−1],[5],[2]],[[2],[1],[−3],[6],[1]]}

Solution

1. Set up a vector equation to check for linear independence by setting a linear combination of the vectors equal to the zero vector.

(c_1)*[[2],[−1],[3],[1],[2]]+(c_2)*[[1],[2],[−1],[5],[2]]+(c_3)*[[2],[1],[−3],[6],[1]]=[[0],[0],[0],[0],[0]]

2. Construct the augmented matrix representing this system of linear equations.

[[2,1,2,0],[−1,2,1,0],[3,−1,−3,0],[1,5,6,0],[2,2,1,0]]

3. Perform row reduction to find the row echelon form of the matrix.

[[1,5,6,0],[0,1,1,0],[0,0,1,0],[0,0,0,0],[0,0,0,0]]

4. Analyze the pivots in the reduced matrix to determine if there are free variables.

Since there is a pivot in every column, the only solution is *(c_1)=(c_2)=(c_3)=0

5. Conclude based on the definition of linear independence.

Because the only solution is the trivial solution, the set is linearly independent.

Final Answer

S=Linearly Independent

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