Determinats,dependence, span, basis.

[am_knightmare]

Im having trouble under stand the relationships between determinats, span, basis.
Given a 3x3 matrix on R3 vector space.
* If determinat is 0, it is linearly dependent, will NOT span R3, is NOT a basis of R3.
, If determinant is non-zero, its linearly independent, will span R3, is a basis of R3

I was not able to confirm this statement one of my friends said, and i checked wiki as well, but didn't find a answer. Question is, is this right? can anyone confirm this? Thanks in advance.

 

[Erland]

It is not meaningful to say that a determinant is linealy independent, spans Rn, or is a basis for Rn. Only sets of sets of vectors can do these things. And the set of vectors of interest here is the set of column vectors of the determinant (more precicely, of the underlying matrix).

For a (quadratic) nxn-matrix A, the following four statements are equivalent, either all four of them are true or all four are false:

1. The determinant of A is nonzero.
2. The column vectors of A is a linearly independent set.
4. The column vectors of A span Rn.
5. The column vectors of A is a basis for Rn.

The corresponding is true for the row vectors.

 

[Alpha Floor]

I think this link might be useful to you, here I answer some of your questions...

"www.physicsforums.com/showthread.php?t=590440"

For now I will say you're mixing up concepts. A determinant itself has nothing to do with linear dependence or with basis... a determinant is simply a number which is assigned to every quadratic matrix. If you understand what a determinant is, then you will be able to understand what the "rank of a matrix" is, which is a number assigned to every matrix (quadratic or not).

I hope I don't get banned or anything for helping you, by the way...

 

[am_knightmare]

Thanks for the replies, just what I needed Erland.

 

[blue_raver22]

Yes, the statement is correct. The determinant of a matrix plays a crucial role in determining its linear independence and whether or not it spans the vector space it is defined on. If the determinant is 0, it means that the matrix is not invertible and therefore, its columns are linearly dependent. This also means that the vectors represented by the columns do not span the entire vector space. In this case, the matrix cannot be a basis for the vector space.

On the other hand, if the determinant is non-zero, it means that the matrix is invertible and its columns are linearly independent. This also means that the vectors represented by the columns span the entire vector space. In this case, the matrix can be considered as a basis for the vector space.

In summary, the determinant of a matrix is a key factor in determining its linear independence, span, and whether or not it can be a basis for a vector space. I hope this helps clarify the relationships between determinants, span, and basis.