Re: Linear algebra with ...T


"quasi" <quasi@xxxxxxxx> wrote in message
news:rkuli31cjknjc9dmdp539qu2c7ilqdk0ep@xxxxxxxxxx
On Thu, 01 Nov 2007 22:23:47 -0700, mina_world@xxxxxxxxxxx wrote:

On 11 2 , 2 07 , William Elliot <ma...@xxxxxxxxxxxxxxxxxx> wrote:
On Fri, 2 Nov 2007, mina_world wrote:
A, B are 4x4 matrix.

Let (A^3)B - 2AB + 3E = 0.

What's E about?

E = I = identity matrix.

T : R^4 -> R^4, T(X) = (2A - A^3)X (X in R^4)

T(B) = 3E.

Find the dimension of T(R^4).

Depends upon A. For example,
0 if A = (sqr 2)I, 4 if A = I.

If B is nonsingular matrix, I proved that T(R^4) = 4.

(A^3)B - 2AB + 3E = 0

=> (2A - A^3) B = 3E

=> det (2A - A^3) det(B) = 81

=> 2A - A^3 and B are both nonsingular

=> im(2A - A^3) = R^4

Oh, clever idea.
Thank you very much.


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