Question

to determine whether the given vectors u,v, and w are linearly independent or dependent. If they are linearly dependent, find scalars a,b, and c not all zero such that au+bv+cw=0. 19. u=(2,0,1),v=(−3,1,−1),w=(0,−2,−1)

Answer 1

The vectors u, v, and w are linearly independent.

Step-by-step explanation:

Linear Dependence or Independence of Vectors

To determine whether the given vectors u, v, and w are linearly independent or dependent, we can create an augmented matrix with the vectors as columns. If the rank of the matrix is less than the number of vectors (3 in this case), then the vectors are linearly dependent. If the rank is equal to the number of vectors, then the vectors are linearly independent.

Let's create the augmented matrix for the given vectors:

| 2 -3 0 || 0 1 -2 || 1 -1 -1 |

Now we can perform row operations to determine the rank of the matrix:

R2 = R2 + 1/2 R1R3 = R3 - R1

The matrix becomes:

| 2 -3 0 || 0 1 -2 || 0 2 -1 |

The rank of the matrix is 3, which is equal to the number of vectors. Therefore, the vectors u, v, and w are linearly independent.