Find the coordinate vector of w relative to the basis S=(u1,u2) for R^2.

Question

u1=(2,-5),u2=(4,9);w=(1,1)

(w)s=?

Answer 1

We must find real numbers c₁ and c₂ so that w = c₁u₁ + c₂u_2.

So, we have the system of equations:

2c₁ + 4c₂ = 1

-5c₁ + 9c₂ = 1

Solve for c₁ and c₂ to get c₁ = 5/38 and c₂ = 7/38.

The coordinate vector of w with respect to the given basis is then <c₁, c₂> = <5/38, 7/38>

In other words, <1, 1> = (5/38)<2, -5> + (7/38)<4, 9>