Question

Find a basis of the subspace of R3 of the solution of the equation X+Y+Z=0 .

Answer 1

Answer on Question #40301, Math, Linear Algebra

Find a basis of the subspace of R3 of the solution of the equation $X + Y + Z = 0$.

**Solution.**

To find basis, we need to convert this equation into a vector equation.

Solve the equation for $x$.

$$x = -y - z.$$

Now let $y$ and $z$ be parameters.

Letting $y = s$ and $z = t$, we find that any vector $(x, y, z)$ that lies in the plane can be written as vector equation

$$(x, y, z) = (-s - t, s, t)$$

for all $s$ and $t$.

Separating the parameters, we find that

$$(x, y, z) = s(-1, 1, 0) + t(-1, 0, 1)$$

for all $s$ and $t$.

Therefore, the set $\{(-1,1,0), (-1,0,1)\}$ is a basis for the plane $X + Y + Z = 0$.

Answer: $\{(-1,1,0), (-1,0,1)\}$