Rows linearly independent implies columns linearly independent

Question

If you have an $m \times n$ matrix and the rows are linearly independent, are the columns linearly independent too?

I know that it's the case for $n \times n$ matrices.

I've yet tried to think about it, and I think the answer is yes, for example the matrix: $\begin{bmatrix} 1 & 2 & 3 \\ a & 2a & 3c \end{bmatrix}$ has linearly dependent rows and columns, but I'm not sure that it works for every $m \times n$ matrix.

Say you have $m$ rows and $n$ columns with $n>m.$ You can never have $n$ linearly independent $m$-dimensional vectors. — saulspatz, Jun 08 '19 at 13:34
Does this answer your question? How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$? — Jyrki Lahtonen, Nov 01 '20 at 15:50

Tesla · Accepted Answer · 2019-06-08T13:29:05.110

3

Take

$\begin{pmatrix}1 &1 &0 \\2 &0 &1\end{pmatrix}$

then the rows are linear independent, but the columns aren't, since for example for the first column you have$$\begin{pmatrix} 1 \\ 2\end{pmatrix}=\begin{pmatrix}1 \\0\end{pmatrix}+2\begin{pmatrix}0 \\ 1\end{pmatrix}$$

edited Jun 08 '19 at 13:29

answered Jun 08 '19 at 13:22

Tesla

1,380

thanks a lot ;) – Mari3 Jun 08 '19 at 13:29

score 2 · Answer 2 · answered Jun 08 '19 at 13:36

2

If you have an m x n matrix and the rows are linearly independent, are the columns linearly independent too?

This is only necessarily true when the matrix is square.

answered Jun 08 '19 at 13:36

score 2 · Answer 3 · answered Jun 08 '19 at 15:30

In $m\times n$ matrix, the maximum number of independent rows or columns possible is the order of the largest square you can get from it. If $m >n$ then order of the largest square is n, so you can get at most n linearly independent rows or columns (and vice versa). If you get x linearly independent rows then you will also get same number of linearly independent columns also (and vice versa), even for a rectangular matrix.

score 1 · Answer 4 · answered Jun 08 '19 at 13:32

1

Your set of vectors that form the matrix are linearly independent, iff your matrix is invertible.
only square matrices are invertible.
Therefore, only square matrices are linearly Independent.

answered Jun 08 '19 at 13:32

user679128

226

This is not necessarily a good reasoning. Set that forms a matrix can be linearly independent even in rectangular matrices. The first point is wrong. – Snifkes Dec 14 '21 at 15:13

Rows linearly independent implies columns linearly independent

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