Express v = (7, 1) as a linear combination of u1 = (1, 1) and u2 = (2, -1).

Step 1: Set up the linear combination.

We want scalars a and b such that:

a(1, 1) + b(2, -1) = (7, 1)

Step 2: Match corresponding components.

From the first components:
a + 2b = 7

From the second components:
a - b = 1

Step 3: Solve the second equation for a.

a - b = 1

a = 1 + b

Step 4: Substitute into the first equation.

(1 + b) + 2b = 7

1 + 3b = 7

3b = 6

b = 2

Step 5: Find a.

a = 1 + b

a = 1 + 2

a = 3

Step 6: Write the linear combination.

v = 3u1 + 2u2

Check:
3(1, 1) + 2(2, -1) = (3, 3) + (4, -2) = (7, 1)

Therefore, v = 3u1 + 2u2.