One-to-one Proof

Date: 06/25/98 at 05:49:32
From: will
Subject: One-to-one

How can I do the following proof?
 
Let T(x) = Ax, where A is a 2 x 2 matrix. Show that T is one-to-one if 
and only if the determinant of A is not zero.

Date: 06/27/98 at 14:03:13
From: Doctor Joe
Subject: Re: one-to-one

Dear Will,

I shall assume that you know the standard theorems:

(A) n x n matrix has determinant not zero if and only if the matrix 
    has row rank = n.

(B) For an n x n matrix A, row rank(A) + nullity (A) = n.

Along the way, I shall leave some work to you. This means I shall 
start the proof and leave enough hints for you to go on.  

Now let's start to prove the statement.

(=>)  If the map T is one-one.  Suppose that the det A = 0.

Take any v from Ker A (which is defined as the subspace of R^2 which 
is mapped to the 0-vector of R^2).

Then, T(v) = 0.

But T(0) = 0. This implies that T(v) = T(0).

Exercise: Make use of the fact that T is one-to-one and invoke Theorem 
(A) to deduce the nullity(A). By using Theorem (B), complete the 
proof.

(<=)  Assume that det A is not zero. Then we know that the nullity (A) 
is zero, by Theorem (A) and (B) by the same reasoning. Therefore 
Ker(A) = {0}.

Take any two vectors u and v such that T(u) = T(v).

Therefore, T(u)-T(v) = 0.

Exercise: Use the linear property of T to show that u-v is an element 
of the Ker A. Show that T is one-one.

I hope this helps.

- Doctor Joe, The Math Forum
  http://mathforum.org/dr.math/