Answer:
- x as a linear combination :
x = -1 v1+ 0 v2+ 1 v3.
- Transpose Ax = (12, -6, -6)
Step-by-step explanation:
Given v1 = (0 3 3),v2 = (1 −1 0), v3 = (3 0 −3) be eigenvectors of the matrix A which correspond to the eigenvalues λ1 = −1, λ2 = 0, and λ3 = 1, respectively, and let x = (−2 −4 0). Express x as a linear combination of v1, v2, and v3, and find Ax .
To write x as a linear combination of v1, v2, and v3
x = -1 v1+ 0 v2+ 1 v3.
To find Ax
Write A = (0 ......3 ......3 )
...................(1 ......-1 ......0)
...................(3 ......0......-3)
Since transpose x = (-2, 4, 0)
Ax =......... (0 ......3......3 )(-2)
...................(1 ......-1 ......0)(4)
...................(3 ......0......-3)(0)
= (0×-2 + 3×4 + 3×0)
...(1×-2 + -1×4 + 0×0)
.. (3×-2 + 0×4 + -3×0)
As = (12)
....(-6)
....(-6)
Transpose Ax = (12, -6, -6)
