Exercise Express the vectors b = (4, 6), c = (1, 1), e1 = (1, 0), e2 = (0, 1) as linear combinations of the vectors u = (1, 2), v = (3, 4), w = (5, 6). Can you express as linear combinations of the vectors u, v only?
Answer We would like to write b, c , e1, e2 in the form x1(1, 2) + x2(3, 4) + x3(5, 6). From the previous exercise, we have
|x1[||1||] + x2[||3||] + x3[||5||] = [||x1 + 3x2 + 5x3||]|
|2||4||6||2x1 + 4x2 + 6x3|
Therefore the problem is to solve the system of equations
|x1||+ 3x2||+ 5x3||=||b1|
|2x1||+ 4x2||+ 6x3||=||b2|
For b1 = 4, b2 = 6 on the right side, we find x1 = 1, x2 = 1 is one solution. Therefore b = u + v. In fact, the solution is not unique. For example, x1 = 2, x2 = -1, x3 = 1 is another solution, from which we get b = 2u - v + w.
Incidentally, we have already expressed b as a linear combination of u and v: b = u + v. Such expressions are equivalent to solutions of
We do the similar computation for the other vectors and find c = -1/2 u + 1/2 v, e1 = -2u + v, e2 = 3/2 u - 1/2 v.