GR0568 36

Forum for the GRE subject test in mathematics.
Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

GR0568 36

Postby Hom » Mon Oct 10, 2011 10:20 pm

Let M be a 5 x 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 x 1 column vector b.
(D) The determinant of M is nonzero.
(E) There exists a 5 x 5 real matrix N such that NM is the 5 x 5 identity matrix.

Answer:A

I don't have problem with B/C/D/E. But I don't know about A.
It should be right to say u,v are linearly independent. but what's the set{u,v} about?

mjmiller2011
Posts: 7
Joined: Mon Jul 18, 2011 10:58 pm

Re: GR0568 36

Postby mjmiller2011 » Mon Oct 10, 2011 11:25 pm

The last four answers are equivalent to saying that the matrix is invertible. The first one is not. Lets think of M as a 3x3 matrix, instead of a 5x5 for simplicity's sake. Let the left column read [0 1 1], the middle column read [0 0 1] and the right column read [0 1 0]. These column vectors are linearly independent, but the determinant of the matrix is zero.

blitzer6266
Posts: 61
Joined: Sun Apr 04, 2010 1:08 pm

Re: GR0568 36

Postby blitzer6266 » Tue Oct 11, 2011 12:04 am

I think you meant this mjmiller2011, but it's important to mention that these column vectors are pairwise[u][/u] linearly independent, but not linearly independent as a set of 3 vectors.

Hom
Posts: 39
Joined: Sat Oct 01, 2011 3:22 am

Re: GR0568 36

Postby Hom » Tue Oct 11, 2011 7:20 am

mjmiller2011 wrote:The last four answers are equivalent to saying that the matrix is invertible. The first one is not. Lets think of M as a 3x3 matrix, instead of a 5x5 for simplicity's sake. Let the left column read [0 1 1], the middle column read [0 0 1] and the right column read [0 1 0]. These column vectors are linearly independent, but the determinant of the matrix is zero.


Thank you both. That's great example. And thank you for mentioning the key point.

mjmiller2011
Posts: 7
Joined: Mon Jul 18, 2011 10:58 pm

Re: GR0568 36

Postby mjmiller2011 » Tue Oct 11, 2011 7:34 am

Yeah that's what I meant to say, my bad for not being clear. In my example the set of three vectors is not linearly independent but any two vectors of the set are pairwise linearly independent.




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