Finding a basis for the subsapces

Question

Trying to find the basis of these vector subspaces but I'm a little puzzled as to what I am trying to do. I've turned each of the equations for the subsapce into a matrix and tried to reduce it; I'm not sure how to read the basis off of it from that. Any help would be greatly appreciated. I've attached the question and my workings. Thank you very much!

Answer 1

For $S_1$ you have found, correctly, that a vector $(x,y,z)^T$ of $S_1$ is such that: $$ \begin{pmatrix} x\\y\\z \end{pmatrix}= \begin{pmatrix} 2y-z\\y\\z \end{pmatrix}= y\begin{pmatrix} 2\\1\\0 \end{pmatrix}+ z\begin{pmatrix} -1\\0\\1 \end{pmatrix} $$ so a basis for $S_1$ is the couple of vectors $\{(2,1,0)^T,(-1,0,1)^T\}$. Note that $S_1$ is the plane passing thorough the origin and orthogonal to the vector $(1,-2,1)$ , so it has dimension $2$, and that $(0,0,0)^T$ is an element of any vector subspace of $\mathbb{R}^3$.

For $S_2$ note that it is the intersection of two non parallel planes , so it is a straight line ( dimension $1$): $$ \begin{cases} x-2y+z=0\\ x-z=0 \end{cases} $$ from the second equation we have $x=z$ and, substituting in the first equation, we find $2y=2z \iff y=z$, so the line has the equation: $$ \begin{pmatrix} x\\y\\z \end{pmatrix} =z\begin{pmatrix} 1\\1\\1 \end{pmatrix} $$ and the vector $(1,1,1)^T$ is a basis for $S_2$,