Hi, I need to prove a very basic equality which states that
if a=b , then x*a = x*b
note that a,b, and x can be scalars, matrices or vectors.
I guess I have to prove it using the ring (R, +, *), any help would be appreciated. Thanks!
Hi, I need to prove a very basic equality which states that
if a=b , then x*a = x*b
note that a,b, and x can be scalars, matrices or vectors.
I guess I have to prove it using the ring (R, +, *), any help would be appreciated. Thanks!
This is true for any theory with equality and a binary operation *. That is, this is a corollary of equality axioms as opposed to axioms describing any particular theory, such as theory of groups, rings or vector fields. Equality axioms are usually considered the common basis of all theories, along with such axioms as "A and (A -> B) imply B."
Note that scalars (e.g., real numbers) and matrices do indeed form a ring, but vectors do not because of multiplication. The dot product maps two vectors into a number, not a vector, and the cross product is not associative, nor does it have a multiplicative identity. Vectors are usually considered to be a vector space over a field or its generalization, a module over a ring. In this case, multiplication takes two objects of different type: a scalar and a vector.
Hi, emakarov.
Thanks for your answer.
As I understand, you are saying that this is an axiom, which needs no proof. That is exactly what I think. But apparently my professor does not think so, so he gave it as a homework
So, any ideas how to prove it?
I don't know how to prove it except for the following. Equality axioms may differ from source to source, but one approach is as follows. There are two axioms about equality: (1) for all x, x = x and (2) for all x, y, A[x] and x = y imply A[y] for any formula A. So, we have x * a = x * a by (1) and then replace one occurrence of a with b by (2) to get x * a = x * b.