Let $A$ be a linear map from the vector space $X$ to $Y$ and $T$ be a subspace of $X$ . I want to understand what is the meaning of saying that the restriction $A_{|T}: \to A(T)$ is invertible. Could anybody explain me what is this restriction map?
Thanks for the help.
One way of thinking about it is that if $t$ and $t'$ are in $T$, then $A(t)=A(t')$ implies that $t=t'$.
The map $A_{|T}$ is only defined on $T$, but everywhere on $T$, it goes the same thing as $A$. If $t\in T$, then $A_{|T}(t)=A(t)$.
Here is a familiar example, though not in a linear algebra setting. Let $f(x)=x^2$, for the set $A$ of all real numbers. Then $f(x)$ is not invertible, because for example (-2)^2=2^2$.
But if we restrict $x$ to the set $T$ of non-negative reals, the restriction of $f$ to $T$, which we could call $f_{|T}$, is invertible. If for example we know that $f_{|T}(x)=9$, then we know that $x=3$.
This applies to a function between sets in general, not only vector spaces.
Given $f:X \rightarrow Y$ and a subset $T \subset X$, the restriction $f_{|T}:T \rightarrow Y$ is by definition $f_{|T}(x) = f(x)$. That is, the function is the same, except that its domain is restricted.
If $f$ is a linear map (or any kind of homomorphism), its restriction will also be one by inheritance.
When we restrict a function to a subset of the domain it means we are interested now in what the function is doing to that subset. We don't care what $A(x)$ is when $x\notin T$.
To say that it is invertible is to say that even if $A$ itself is not invertible as a function, when we only consider what $A$ does to the elements of $T$, we can invert the function.
Namely, there exists $B\colon A(T)\to T$ such that $B(A(t))=t$ for every $t\in T$. This condition may be falsified when we consider $A$'s action on $X$ itself, though, but this is not what we came for.
