Question

Let V be the set of all vectors of the form (x1, x2, x3) in R

3

(a) x1 − 3x2 + 2x3 = 0.

(b) 3x1 − 2x2 + x3 = 0 and 4x1 + 5x2 = 0.

Find the dimension and basis for V.

Answer 1

$v :={(x_1,x_2,x_3) \in R^3: 3x_1-3x_2+x_3=0}\\ basis\\ \begin{bmatrix} 1 \\ 0\\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ -3\\ 0 \end{bmatrix},\begin{bmatrix} 0\\ 0\\ 2 \end{bmatrix}\\ dimension=3\\ b) 3x_1-2x_2+x_3=0\\ 4x_1+5x_2=0\\ x_2=\frac{-4}{5}x_1\\ 3x_1-2. \frac{-4}{5}x_1 +x_3=0\\ 3x_1+\frac{8}{5}x_1 +x_3=0\\ \frac{23}{5}x_1 +x_3=0\\ x_3=\frac{-23}{5}\\ basis \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix},\begin{bmatrix} 0\\ \frac{-4}{5}\\ 0 \end{bmatrix},\begin{bmatrix} 0\\ 0\\ \frac{-23}{5} \end{bmatrix}\\ dimension=3$