Final answer:
The vectors v1 = [0 0 -3], v2 = [0 -3 9], v3 = [4 -2 -6] are linearly independent because the determinant of the matrix formed by these vectors is non-zero. Hence, they do span R³.
Step-by-step explanation:
To answer whether the vectors v1 = [0 0 -3], v2 = [0 -3 9], v3 = [4 -2 -6] span R³ (three-dimensional space), we need to check if these vectors are linearly independent. One way to do this is to set up a matrix with these vectors as columns and then find the determinant.
The matrix A formed by vectors v1, v2, and v3 is:
A = |0 0 4|
|0 -3 -2|
|-3 9 -6|
To find the determinant of A, we use the formula det(A) = a(ei - fh) - b(di - fg) + c(dh - eg), where a, b, c, d, e, f, g, h, i are the entries of the matrix. Here, a = 0, b = 0, e = -3, f = 9, c = 4, g = -3, h = -2 and i = -6.
Det(A) = 0(-3(-6) - 9(-2)) - 0(0 - (-3)(-6)) + 4(0 - (-3)(-2)) = 0 + 0 + 24 = 24
Since the determinant of A is non-zero, the matrix is invertible, which means vectors v1, v2, and v3 are linearly independent and therefore span R³.