Question

Let W={(X,Y,Z)|x+y+z=0}

A, give a basis for W

B, what is the dimension of W

Answer 1

Part A

$x+y+z=0\>$

$\implies\>x=-y-z$

$\begin{pmatrix} -y&-z \\ y \\ z \end{pmatrix}$ =$y\begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}$ +$z\begin{pmatrix} -1 \\ 0\\ 1 \end{pmatrix}$

rref of $\begin{pmatrix} -1 & -1 \\ 1 & 0\\ 0&1 \end{pmatrix}$ =$\begin{pmatrix} 1& 0 \\ 0 & 1\\ 0&0 \end{pmatrix}$

Basis are [$\begin{pmatrix} -1 \\ 1\\ 0 \end{pmatrix}$ , $\begin{pmatrix} -1 \\ 0\\ 1 \end{pmatrix}$ ]

Part 2

Number of linearly independent columns in $W$ is 2

$\therefore$ The dimension of W =2