$($a$)$ Prove that

$H=\{([x],[y],[z])in{R}^3$:$x+y+z=0\}$

is a subspace of $R^3.$

$($b$)$ Write down a set of five linearly independent vectors in $R^4,$ or explain why it is

impossible to do so$.$

$($c$)$ Consider the matrix

$A=([1,-1,5,2,0],[0,1,-4,-2,1],[0,0,0,1,6],[0,0,0,-1,-6])$

$($i$)$ Write down a basis for the row space of $A,$ and determine the rank of $A.$

$($ii$)$ Using your answer to part $($i$),$ determine the nullity of $A.$