Questions: Testing for Linear Independence In Exercises 29-40, determine whether the set S is linearly independent or linearly dependent. 39. S=(4,-3,6,2),(1,8,3,1),(3,-2,-1,0)

Solution Steps

To determine if the vectors are linearly independent, we need to check if the only solution to the equation $c_1v_1 + c_2v_2 + c_3v_3 = 0$ is $c_1 = c_2 = c_3 = 0$, where $v_1 = (4, -3, 6, 2)$, $v_2 = (1, 8, 3, 1)$, and $v_3 = (3, -2, -1, 0)$.

The equation $c_1v_1 + c_2v_2 + c_3v_3 = 0$ can be written as a system of linear equations: $4c_1 + c_2 + 3c_3 = 0$ $-3c_1 + 8c_2 - 2c_3 = 0$ $6c_1 + 3c_2 - c_3 = 0$ $2c_1 + c_2 + 0c_3 = 0$

We can represent this system with an augmented matrix:

[ 4  1  3 | 0 ]
[-3  8 -2 | 0 ]
[ 6  3 -1 | 0 ]
[ 2  1  0 | 0 ]

Using Gaussian elimination (or any preferred method for solving systems of linear equations) on the augmented matrix, we will eventually find that it reduces to:

[ 1  0  0 | 0 ]
[ 0  1  0 | 0 ]
[ 0  0  1 | 0 ]
[ 0  0  0 | 0 ]

Final Answer: The set S is linearly independent