Are the vectors u 3 3 2 v 0 1 1 and w 4 4 2 linearly inde

Are the vectors u = [-3 3 -2], v = [0 1 1] and w = [-4 4 2] linearly independent? If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are linearly independent, find the only scalars that will make the equation below true.

Solution

The matrix from of given vectors:

Transform matrix to the reduced row echelon form :

Step 1: Multiply the 1st row by -1/3:

Step 2: Add -3 times the 1st row to the 2nd row and add 2 times the 1st row to the 3rd row:

Step 3: Add -1 times the 2nd row to the 3rd row:

Step 4: Multiply the 3rd row by 3/14 and add -4/3 times the 3rd row to the 1st row:

which corresponds to the system

a = 0, b = 0 and c = 0

Therefore the set S = {u, v, w} is linearly independent.

-3 0 -4 3 1 4 -2 1 2