Final answer:
To show that the collection of vectors S = {(1,2,3),(0,1,1),(-1,0,-1)} forms a basis for R³, we need to verify two conditions: linear independence and spanning. By solving a system of equations and checking the determinant of a matrix, we can show that the vectors are linearly independent and span R³.
Step-by-step explanation:
To show that the collection of vectors S = {(1,2,3),(0,1,1),(-1,0,-1)} forms a basis for R³, we need to verify two conditions:
1. The vectors in S are linearly independent.
2. The vectors in S span R³.
To check for linear independence, we set up the equation a(1,2,3) + b(0,1,1) + c(-1,0,-1) = (0,0,0), where a, b, and c are scalars. This equation can be rewritten as the following system of equations:
a - c = 0
2a + b = 0
3a + b - c = 0
Solving this system of equations, we find that a = 0, b = 0, and c = 0. Therefore, the vectors in S are linearly independent.
To check if the vectors span R³, we can create a 3x3 matrix A using the vectors in S as columns. Taking the determinant of A, we find that det(A) = 4, which is non-zero. This means that the vectors in S span R³.
Since the vectors in S are both linearly independent and span R³, we can conclude that the collection of vectors S forms a basis for R³.