Question

Consider the real space R³

 The following vectors form a basis S of R^3:

u1 = (1, −1, 0), u2 = (1, 1, 0), u3 = (0, 1, 1)

Find the coordinate vector [v] of v = (5, 3, 4) relative to the basis S .

Answer 1

Let

$$\vec v=a\vec{u_1}+b\vec{u_2}+c\vec{u_3}$$

Then

$$a+b=5$$

$$-a+b+c=3$$

$$c=4$$

We have

$$a+b=5$$

$$2b+c=8$$

$$c=4$$

Then

$$a=3$$

$$b=2$$

$$c=4$$

Vector $\vec{v}$ has the coordinates $\langle 3,2,4\rangle$ relative to the basis S.