Question About equal signs in Theorems

Question

Lets say for example, $L$ is a linear mapping with $V$ as its domain and $W$ as its co-domain.

Theorem: $L(0) = 0$. (Zero vectors)

I know its wrong to assume that every zero vector in $W$ is mapped from a zero vector in $V$, but I can't logically explain why!

Since $L(0) = 0$, and equal signs go both ways (double implication), isn't the statement $L(0) = 0$ implying the $0$ vector in $V$ only maps to the zero vector in $W$, and the zero vector in $W$ can be only mapped from the zero vector in $V$?

I know this is wrong. Can someone explain why?

Thanks

Answer 1

$L(0)=0$ indeed means that $0$ maps to $0$ by the linear transform $L$.

But this doesn't tell you anything about $L(v)$ for other $v$, which might be such that $L(v)=0$ ($L$ might be non-invertible).

Answer 2

Let us start with an example. Let $A$ be a linear map from $$A:R^2 \to R$$ defined as $A(x,y)=x$. $$$$ Now clearly $A(0,0)=0$, but $A(0,1)=0$ and $(0,1) \not= (0,0)$. Now using the definition of the linear map $$0=1-1=A(1,x)-A(1,y)=A((1-1),x-y)=A(0,x-y)$$ if $x\not=y$ then the zero vector in $R$ does not map only to the zero vector in $R^2$. Now the above example shows why "the zero vector in W can be only mapped from the zero vector in V" is incorrect. If you understand what the linear map is doing you can see that in fact the entire $y-axis$ in $R^2$ can be mapped to zero in $R$ by setting $x=0$ in $(x,y)$. $$$$ In the two seperate spaces $V$ and $W$ the zero elements are unique respectively. Now the zero vector in $V$ may not be the only element in $V$ that maps to the zero vector in $W$. You will need more info about the linear map $L$ (one to one, onto, etc.) to determine whether the zero vector in V is the only element in V that maps to the zero vector in W.

Answer 3

The equal sign goes both ways in the sense that $L(0) = 0$ is saying that the vectors $L(0)$ and $0$ are the same vectors in $W$.

If you are trying to view it as an implication, it is indeed a one-way implication saying that "if $v = 0$ then $L(v) = 0$." It does not mean however that the zero vector in $V$ is the only vector $v$ for which $L(v)=0$. To say that we would need to say that "$L(v) = 0$ if and only if $v=0$", or that "$0 \in V$ is the preimage of $0 \in W$ under $L$".