linear algebra - Find a basis of the subspaces of $mathbb{R}^4$ generated by the vectors

Find a basis of the subspaces of $\mathbb{R}^4$ generated by the vectors $$v_1=(1,1,2,0),v_2=(-1,0,1,0),v_3=(2,-2,0,0),v_4=(0,0,-1,2)$$

First of all I wrote these vectors as rows of a matrix then applied the following transformations to reduce the matrix in row echelon form:-
$R_2+R_1$ & $R_3-2R_1$

Then $R_3+4R_2$ and finally $\frac{1}{10}R_3$
Then I wrote the non zero rows in row echelon form as $B={(1,1,2,0),(0,1,3,0),(0,0,1,0)}$ which forms basis .
Am I right here?

Answer

Using the vectors given in the problem, we can define a set: $$A = \{v_1, v_2, v_3, v_4\}$$

Remember the definition of a Basis set:

Given: V is a vector space and B is a subset of V, we say that B is a basis of V IFF $V = span(B)$ and B is linearly independent.

Row-Reducing the matrix form of the set A yields:
$$\begin{bmatrix}1&-1&2&0\\1&0&-2&0\\2&1&0&-1\\0&0&0&2\end{bmatrix}$$
$$ <=>$$
$$\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$$
Since each column contains a leading "1", we can say that the set A is linearly independent.

This result implies that the set A itself satisfies the definition of a basis.

$B = \{v_1,v_2,v_3,v_4\}$