Question

Complete the set {(1,0,2) , (2,3,1) } to get a basis of R3.

Answer 1

Answer on Question #40011, Math, Linear Algebra

Complete the set $\{(1,0,2), (2,3,1), (\ldots)\}$ to get a basis of $R^3$.

Solution.

$$v_1 = (1,0,2)$$ $$v_2 = (2,3,1)$$ $$v_3 = (x, y, z)$$

The best way to guarantee that the vectors are basis is to proof that vectors are orthogonal:

Orthogonal means the dot product $v1 \cdot v2 = 0$.

$$v_1 \cdot v_3 = x + 2z = 0$$ $$v_2 \cdot v_3 = 2x + 3y + z = 0$$

So you can go ahead and make a choice -- you have two equations, three unknowns, so you're allowed to make a choice. Say $x = 1$.

Then

$$z = -2$$ $$y = 0$$

Thus,

$$v_3 = (1, -2, 0)$$

Answer:

$$v_3 = (1, -2, 0)$$