> with(linalg[matrix]):

(2.1.15) Let u₁ = (1, 2, 3) and u₂ = (3, 2, 1).

(a) Express w= (4, 4, 4) as a linear combination of u₁ and u₂.

A quick glance at the vectors reveals that wis just the sum of u₁ and u₂.

> u[1] = [1, 2, 3]; u[2] = [3, 2, 1]; w = u[1] + u[2]; subs({%%, %%%}, %);

(b) Find an example of a vector in R³that is notequal to a linear combination of u₁ and u₂.

One vector that isn't a linear combination of u₁ and u₂ is their cross product; this vector is perpendicular to both, so it can't be expressed as the sum of multiples of u₁ and u₂.

> v = cross([1, 2, 3], [3, 2, 1]);