Hi,

Tell whether or not the following vectors are linear independent, if they are l.i., say if they generate $\displaystyle \mathbb{R}^3 $ or $\displaystyle \mathbb{R}^4$ and in the contrary case characterize implicitly the subspace generated and give a basis of this subspace.

The vectors are (1,1,2,4),(2,-1,-5,2),(1,-1,-4,0) and (2,1,1,6).

My attempt : I formed and reduced this matrix : $\displaystyle \begin{bmatrix} 1&2&1&2 \\ 1&-1&-1&1 \\ 2&-5&-4&1 \\ 4&2&0&6 \end{bmatrix}$ to this one $\displaystyle \begin{bmatrix} 1&0&-\frac{1}{3}&\frac{4}{3} \\ 0&1&\frac{2}{3}&\frac{1}{3} \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix}$. I concluded by saying that the vectors are linear dependent (because of the 2 rows that are 0, so 2 of the 4 vectors are linear dependent between them) and that the vectors generate $\displaystyle \mathbb{R}^2$ (because 2 rows are reduced) and that a basis of $\displaystyle \mathbb{R}^2$ is $\displaystyle \{ (1,0,0,0),(0,1,0,0) \}$.

I have some questions : first, is what I've done correct (at least logically)?

Second question : how can I describe implicitly $\displaystyle \mathbb{R}^2$?

Third question : is $\displaystyle \{ (1,0,0,0),(0,1,0,0) \}$ a possible basis of $\displaystyle \mathbb{R}^{2}$? What about $\displaystyle \{ (1,0,0,0),(0,0,1,0) \}$? (I guess yes).

Thank you very much.