Originally Posted by

**arbolis** Hi MHF,

I know it's a very basic exercise but as I'm unsure about what I've done, I'd like a check up.

In each case, say if the following vectors generates $\displaystyle \mathbb{R}^3$. If not, characterize implicitly the generated subspace.

1)$\displaystyle (1,0,-1),(1,2,1),(0,-3,2)$.

My attempt : I wrote these 3 vectors as column vectors in a matrix and I calculated the determinant of the matrix. I got that it is different from zero, hence the matrix is invertible and the 3 vectors are linear independent. From it I concluded that they generate $\displaystyle \mathbb{R}^3$.

2)(2,-2,0),(2,-2,1),(-1,4,-3).

My attempt : I did the same as in 1) and got exactly the same conclusion.

3)(1,-2,2),(2,1,3),(0,3,1),(1,2,3).

My attempt : Here it gets interesting. The 4 vectors cannot be all linear independent,but they still might generate $\displaystyle \mathbb{R}^3$. I wrote them as vector columns in a matrix and reduced it. I got that the 3x4 matrix has 3 rows reduced. So the 3 first vectors generates $\displaystyle \mathbb{R}^3$.

4)(1,3,-3),(2,3,-4),(1,-3,1),(3,0,-3).

My attempt : I wrote the vectors as column vectors in a matrix and reduced it. I got that a row of the matrix is null. While 2 rows are reduced, hence the 2 first vectors generate $\displaystyle \mathbb{R}^2$.

Am I right? Did I give the generated subspace implicitly in 4)? By the way for 3) and 4) I wrote an amplied matrix with the column a,b,c. I don't think it was of any use now... or was it useful to describe the subspace generated implicitly?