The following is a question from the sample GRE Mathematics Subject Test
found on the ETS website:
Let $M$ be a $5\times 5$ real matrix.
Exactly four of the following five conditions on $M$ are equivalent
to each other.
Which of the five conditions is equivalent to NONE of the other four?
(A) For any two distinct column vectors $u$ and $v$ of $M$,
the set {$u,v$} is linearly independent.
(B) The homogeneous system $Mx=0$ has only the trivial solution.
(C) The system of equations $Mx=b$ has a unique solution for each real $5\times 1$
column vector $b$.
(D) The determinant of $M$ is nonzero.
(E) There exists a $5\times 5$ real matrix $N$ such that $NM$ is the $5\times 5$ identity matrix.
Apparently, the correct answer is (A), but I can't figure out why this is true. If $M$ is
nonsingular, as is implied by statements (B)-(E), then isn't that equivalent to its column vectors being linearly
independent? And if the 5 column vectors are independent, then I can easily show that
each pair of vectors are independent. What am I missing?
Best Answer
Linear independence of $n$ vectors, for $n>2$, is not equivalent to "pairwise" independence. Take three different vectors in the plane, for an example.