1. ## Linear Dependency

I have begun learning about linear dependency in class, but I am confused about a concept. I know that if you have a matrix with more columns than rows it must be linearly dependent, since you will have free variables. However, if you have more rows than columns, and end up with a row full of zeroes, is the matrix linearly independent since there are infinite solutions?

2. Originally Posted by temaire
I have begun learning about linear dependency in class, but I am confused about a concept. I know that if you have a matrix with more columns than rows it must be linearly dependent, since you will have free variables. However, if you have more rows than columns, and end up with a row full of zeroes, is the matrix linearly independent since there are infinite solutions?
If a matrix is invertible, then the column vectors of that matrix (also its row vectors) are linearly independent. But a matrix with more columns than rows is not square and hence not invertible.

3. Originally Posted by Roam
If a matrix is invertible, then the column vectors of that matrix (also its row vectors) are linearly independent. But a matrix with more columns than rows is not square and hence not invertible.
So then a matrix with more rows than columns is also not invertible and thus linearly dependent?

4. Originally Posted by temaire
So then a matrix with more rows than columns is also not invertible and thus linearly dependent?
Yes a matrix whose number of rows and columns are not equal is not square, and therefore it has no inverse.
A good way to check invertiblity of a matrix is to find its determinant. If the determinant is zero every where the system is linearly dependent. If the determinant is ever nonzero the system is linearly independent. (This is because zero determinant means the matrix is not invertible).

Edit: I forgot to mention, every set of "m+1" or more m-dimensional vectors of the same type (either row or column) is linearly dependant. If a set contains more vectors than the dimension of its member vectors (i.e. the number of columns is different from the number of rows) then the vectors in that set/matrix are linearly dependant.