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        riddles >> putnam exam (pure math) >> rank and column space of a matrix
        
(Message started by: Nikki on May 27^th, 2004, 6:48am)

Title: rank and column space of a matrix
Post by Nikki on May 27^th, 2004, 6:48am

a) suppose that A is a 3*3 matrix whose nullspace is a line thorugh the origin in R3. Can the row or column space of A also be a line through the origin? ExplaiN

b) If A is a 6*4 matrix such that the system Ax=0 has a non-trivial solution, what is the largest possible value of rank (A)? explaiN

Title: Re: rank and column space of a matrix
Post by THUDandBLUNDER on May 27^th, 2004, 9:00am

Quote:

explaiN

Your punctuation is rather poor. You should write "Explain!"

Title: Re: rank and column space of a matrix
Post by Sir Col on May 27^th, 2004, 11:10am

on 05/27/04 at 09:00:45, THUDandBLUNDER wrote:

Your punctuation is rather poor. You should write "Explain!"

Your punctuation is almost as poor. You should have written...

Your punctuation is rather poor. You should write, "Explain!" ;D

Nikki, I'm afraid that I don't understand the question. I thought that the null space is the set of vectors, v, that solve the equation, Av = 0. How can a vector be a line through the origin? Perhaps one of our resident experts in Linear Algebra can assist.

Title: Re: rank and column space of a matrix
Post by Icarus on May 27^th, 2004, 7:57pm

One vector cannot be a line. However a "set of vectors" can be a line! The nullspace is presumably exactly the set Sir Col identified - though it is more often called the "kernal". This set is easily seen to be closed under addition and scalar multiplication, so it must be a vector space itself. Therefore it must consist of the origin alone, or a line through the origin, or a plane through the origin, or all of space.

In this case, we are given that it is a line, which means that the matrix A has exactly two linearly independent rows or columns.

This is the best I can tell you though, because the terminology "Row or column space" is not one I have ever heard before. What do you mean by these?

As for (b), the equation is: Rank + dimension of kernal (nullspace) = Dimension of the domain.

Since Ax = 0 has a non-trivial solution, what is the smallest dimension possible of the nullspace?

Title: Re: rank and column space of a matrix
Post by Eigenray on May 27^th, 2004, 11:23pm

The row/column space is the span of the rows/columns. These spaces have the same dimension, which is the rank of the matrix. The dimension of the nullspace is called the nullity.

Nikki, to relate the rank and nullity of a matrix, there is a theorem called, conveniently enough, the "rank-nullity theorem", and for an m x n matrtix, tells you that rank A + nullity A = n.

So, if the nullspace is a line, then its dimension is what? Then what is the rank, i.e., dimension of the row or column space? Can they be lines?

Title: Re: rank and column space of a matrix
Post by THUDandBLUNDER on May 29^th, 2004, 3:24am

Quote:

...though it is more often called the "kernal".

...but not outside of Kansas - 'round these here parts it goes by the name of 'kernel'. :P