Use a graphing utility or computer software program with mat

Use a graphing utility or computer software program with matrix capabilities to write v as a linear combination of u_1, u_2, u_3, u_4, u_5 and u_6. Then verify your solution. u_1 = (4, 3, -1, -1, 1, 1) u_2 = (3, -1, 3, -3, 2, 1) u_3 = (1, 1, -3, -3, -2, 3) u_4 = (2, 2, -1, 4, 1, 2) u_5 = (1, -4, -1, 4, -1, 3) u_6 = (2, 4, -1, 2, 1, 4) v = (22, 57, -10, -15, 12, 5)

Solution

We have used a row ehelon form reducing calculator to compute RREF of both U ( with the given vectors u₁ to u₆ as columns) and (U :V ) with the vectors u₁ to u₆ and v as columns). The RREF of U is I₆ and the RREF of (U:V) is I₆ plus an additional column ( 5, - 1, 1, 2, - 6, 3)^T . Therefore v is a linear combination of the vectors u₁,u₂,u₃,u₄,u₅ and u₆ and v = 5u₁ - u₂ + u₃+ 2u₄ - 6u₅ + 3u₆ We can verify this as under:

i. 20 - 3 + 1+ 4 - 6+6 = 22

ii. 15+1+1+4+24+12 = 57

iii. - 5 - 3 - 3 - 2+ 6 - 3 = -10

iv. - 5 + 3 - 3 + 8 - 24 + 6 = -15

v. 5 - 2 - 2 + 2 + 6 + 3 = 12

v. 5 - 1 + 3 + 4 - 18 + 12 = 5