C1W1: Independent columns and rows of a matrix

[hiddenpieces]

Hello,
In the first week, we learned about the linear dependent/independent rows of a matrix. As I can see, rows are dependent if one row is a linear combination of other rows.

When I learned other concepts for examples: span or spaces. The dependent/independent columns of a matrix are always considered instead. And it turns out that the matrix is represented by many column vectors, which makes the column independence/dependence more reasonable.

I wonder if there is any difference between the independence/dependence of rows and columns.
Is there any case, that a matrix has independent rows and dependent columns simultaneously?
Can we use both to determine whether a system is independent or not? What are the applications of each?

Thank you for your help!

[paulinpaloalto]

If the rows of a matrix are all linearly independent, then the columns of a matrix are also linearly independent. One way to see that is that we have the following relationship:

|A^T| = |A|

That is to say, the determinant of A^T (the transpose of A) is the same as the determinant of A. So that means that if A is non-singular, then A^T is also non-singular. And if A is singular, so is A^T.

Prof Luis does not discuss “span”, but I’m guessing it is the same concept that is called “rank” in this course: it is the number of basis vectors of the linear transformation that the matrix expresses.