Matrix equality multiplication

Question

Assume that we have the equality $x = y$ for $x,y \in R^n$. can one conclude that $Ax = Ay$, for a given matrix A.

If it helps, one can assume that $A$ is positive semi-definite matrix.

Answer 1

If $x=y$, then we have $Ax=Ay$.

Since we are multiplying $A$ with the same vector, we do not need to assume $A$ is positive semi-definite as long as $A \in \mathbb{R}^{m \times n}$.

In general, if $x=y$ and $f$ is a function, then we have $f(x)=f(y)$.

Answer 2

Yes, from $x=y$ we get $Ax=Ay$, by the definition of the matrix-vector product.

Answer 3

Think about it for a second. If $x=y$ that means that $x$ and $y$ are simply two symbols for the same thing. That means you can replace any instance of $x$ with $y$ and vice versa. Hence, obviously you have that $$Ax = Ay $$ Because $Ax$ is simply another way to write $Ay$ because $x$ and $y$ are the same thing!