Question

Find the nullity and the range of the space spanned by 𝑢 = (1,0,2), 𝑣 = (−3,1,1), 𝑤 = (0,1,4) and 𝑥 = (−1, −3,5)

Answer 1

$\begin{bmatrix} 1&-3&0&-1\\ 0&1&1&-3\\2&1&4&5 \end{bmatrix}$

$R_3-2R_1\to\>R_3$

$R_1+3R_2\to\>R_1$

$\begin{pmatrix} 1&&0&&3& &-10\\ 0&&1&&1&&-3\\ 0&&7&&4&&7 \end{pmatrix}$

$\frac{R_3-7R_2}{-3}\to\>R_3$

$\begin{pmatrix} 1&&0&& 3&&-10 \\ 0&&1& & 1&&-3\\ 0&&0&&1&&\frac{-28}{3} \end{pmatrix}$

$R_1-3R_3\to\>R_1$

$R_2-R_3\to\>R_2$

rref is $\begin{bmatrix} 1&0&0& 18 \\ 0&1&0&\frac{19}{3}\\0&0&1&-\frac{28}{3} \end{bmatrix}$

Nullity =1

Range=$[\begin{bmatrix} 1 \\0\\ 2 \end{bmatrix},\begin{bmatrix} -3 \\1\\ 1 \end{bmatrix},\begin{bmatrix} 0\\ 1\\4 \end{bmatrix}]$