Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

$A$ is an $M\times N$ matrix with linearly independent rows and linearly independent columns. Prove that $A$ must be square matrix.

share|cite|improve this question
Assume wlog that $M<N$. Then $A:\Bbb R^N\to \Bbb R^M$ must have nontrivial kernel... – anon Oct 10 '12 at 21:27
You also can see this by the dimensionforumula for functions in $Hom(\mathbb R^n, \mathbb R^m)$. Use this to conclude that $\dim \ker A >0$ with $im(A) = \mathbb R^m$. – Epsilon Oct 10 '12 at 21:48
$\rank(A)=$ the number of linearly independent columns, and $\rank(A)=$the number of linearly independent rows....... – N. S. Oct 10 '12 at 23:07

Hint: The matrix must have full row and column rank, but $\mathrm{rank} (A) \le \min(m,\ n)$

share|cite|improve this answer

Assume $M\leq N$ (otherwise we can work with $A^T$). So we have $N$ columns, each a vector in $\mathbb{R}^M$; the maximum number of linearly independent vectors in $\mathbb{R}^M$ is $M$, so we have $N\leq M$. Then $N=M$.

share|cite|improve this answer

If $V$ is a vector space with $\dim V=k$, and you pick $v_1,\, v_2,\ldots,\, v_p$ in $V$ with $p\gt k$ then those vectors must be l.d. because otherwise we would have a subspace $W$ of $V$ with $\dim W\gt \dim V$, namely $W=\langle v_1,\ldots,v_p\rangle$.

If $A$ is a matrix with real entries and $M\lt N$, the columns of $A$ are $N$ vectors of $\Bbb R^M$ therefore they are l.d. Since this is contradicts that $A$ have linearly independent columns, it must be $M\geq N$.

Now, $A^\text{T}$ is a $N\times M$ matrix with linearly independent rows and linearly independent columns. By what we already have, it follows that $N\geq M$.

Therefore $M=N$.

share|cite|improve this answer
@Justin If you liked this answer or any of the other answers provided by the other users then you can click the check mark beside the answer to "accept" it. If you have 15 rep or above, then you can even upvote an answer to show gratitude; you can upvote multiple answers. This will help the user and yourself. – leo Nov 23 '13 at 16:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.