Does the following vectors generate R^3?
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 . If not, characterize implicitly the generated subspace.
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 .
My attempt : I did the same as in 1) and got exactly the same conclusion.
My attempt : Here it gets interesting. The 4 vectors cannot be all linear independent,but they still might generate . 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 .
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 .
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?