Math Help Forum - View Single Post - linear algebra- vector-basis-subspace

arbolis · #1 October 10th, 2008, 10:53 AM

Hi,
Tell whether or not the following vectors are linear independent, if they are l.i., say if they generate $\mathbb{R}^3$ or $\mathbb{R}^4$ 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 : $\begin{bmatrix} 1&2&1&2 \\ 1&-1&-1&1 \\ 2&-5&-4&1 \\ 4&2&0&6 \end{bmatrix}$ to this one $\begin{bmatrix} 1&0&-\frac{1}{3}&\frac{4}{3} \\ 0&1&\frac{2}{3}&\frac{1}{3} \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix}$ . 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 $\mathbb{R}^2$ (because 2 rows are reduced) and that a basis of $\mathbb{R}^2$ is $\{ (1,0,0,0),(0,1,0,0) \}$ .
I have some questions : first, is what I've done correct (at least logically)?
Second question : how can I describe implicitly $\mathbb{R}^2$ ?
Third question : is $\{ (1,0,0,0),(0,1,0,0) \}$ a possible basis of $\mathbb{R}^{2}$ ? What about $\{ (1,0,0,0),(0,0,1,0) \}$ ? (I guess yes).
Thank you very much.