Linear Algera ?Dependent Rows?

1. Feb 19, 2012

1nvisible

If the rows of A are linearly dependent, are the rows of AB also linearly dependent? Justify answer.

I don't completely understand this question because (so far) my instructor and my text book has not discussed what it means for rows to be linearly dependent. There is a similar question in my text book that asked "if the columns of B are linearly dependent, show that the columns of AB are also."

Does this have something to with transverses? Can I just think about it the same way I have been with columns or does "row" dependencs mean something different?

2. Feb 19, 2012

alanlu

(AB)T = BTAT, so yes, they're more or less the same problem.

Vectors v1, …, vn are linearly independent if k1v1 + … + knvn = 0 (the zero vector) implies the scalars ki must be equal to 0.

They are linearly dependent if one of the vectors can be written as a linear combination of the others (or is contained in their span). This is equivalent to saying they're not linearly independent.

(Sorry for the multiple edits, a bit rusty.)

Last edited: Feb 19, 2012
3. Feb 19, 2012

HallsofIvy

In particular, the rows of matrix A are linearly independent if and only if A is invertible. But then $A^T$ is also invertible and its rows are A's columns.

4. Feb 19, 2012

alanlu

This is true only for square matrices. I believe the question is asking about matrices in more generality.

I would write the components of a row vector in AB in terms of the dot products of a row vector in A and column vectors in B. Which row vector in AB can be written as a linear combination of the other row vectors in AB?