Question

Express the following as linear combinations of u = (2,1,4), v = (1,-1,3), and w = (3,2,5).

(a) (-9,-7,-15)
(b) (6,11,6)
(c) (0,0,0)

Answer 1

To express a vector as a linear combination of u, v, and w, we find coefficients such that the vector is equal to a*u + b*v + c*w. Using this method, we can express vectors (-9,-7,-15), (6,11,6), and (0,0,0) as linear combinations of (2,1,4), (1,-1,3), and (3,2,5).

Step-by-step explanation:

To express a vector as a linear combination of u, v, and w, we need to find coefficients a, b, and c such that the vector is equal to a*u + b*v + c*w.

(a) (-9, -7, -15) = a*(2,1,4) + b*(1,-1,3) + c*(3,2,5)

Solving the system of equations, we find that a = -3, b = -1, and c = 2. Therefore, (-9, -7, -15) = -3*(2,1,4) - (1,-1,3) + 2*(3,2,5).

(b) (6, 11, 6) = a*(2,1,4) + b*(1,-1,3) + c*(3,2,5)

Solving the system of equations, we find that a = 5, b = -3, and c = 0. Therefore, (6, 11, 6) = 5*(2,1,4) - 3*(1,-1,3).

(c) (0, 0, 0) = a*(2,1,4) + b*(1,-1,3) + c*(3,2,5)

Solving the system of equations, we find that a = 0, b = 0, and c = 0. Therefore, (0, 0, 0) = 0*(2,1,4) + 0*(1,-1,3) + 0*(3,2,5).