Can A=B be Proved Algebraically without Linear Transformations?

[Bipolarity]

Suppose A and B are matrices of the same size, and x is a column vector such that the matrix products Ax and Bx are defined.

Suppose that Ax=Bx for all x. Then is it true that A=B?

I know that this is true and I can prove it using the idea of transformation matrices, and viewing Ax and Bx each as linear transformations and showing that those two transformations are equivalent, but I was curious if this can be proved without appealing to the notion of a linear transformation.

Tips?

BiP

 

[Fredrik]

If you view them as linear transformations, then there's nothing to prove, since "Ax=Bx for all x" is by definition what A=B means. (This holds for all functions A and B that have the same domain, not just for the linear ones).

If you don't, then you can do it by trying many different choices of x. For example, if you try (1,0,...,0), then the equality tells you that the first column of A is equal to the first column of B.

 

[lurflurf]

A=B <=> A-B=0
write as
(A-B)x=0

transform way
Tx=0 for all x
T=0

matrix way
the matrix is defined by the action on any basis
Tx=0 for all x
let B be a basis
TB=0
T=0