Question

suppose W is plane 3x +6y-4z=0. a basis for W is?

Answer 1

Writing z in terms of x and y,

$-4z=-3x-6y$

$z=\frac{3}{4}x+\frac{3}{2}y$

Any vector basis,

$(x,y,z)=(x,y,\frac{3}{4}x+\frac{3}{2}y)$

When $x=1,y=0,z=\frac{3}{4}$

Setting $x=0,y=1,z=\frac{3}{2}$

$(x,y,\frac{3}{4}x+\frac{3}{2}y)$

$=x (1,0,\frac{3}{4})+y(0,1,\frac{3}{2})$

Basis is given by

$$\{\begin{bmatrix} 1\\ 0\\¾ \end{bmatrix},\begin{bmatrix} 0\\ 1\\3/2 \end{bmatrix}\}$$