Question about notation in Linear Algebra

[BiGyElLoWhAt]

Given vector spaces V, W, and a function T:V→W , state the two equations that the function T must satisfy to be a linear function.

Does T:V→W mean a function that maps vectors in V into W? Or what does this actually mean?

 

[BruceW]

pretty much, yeah. The domain of the function is V and the codomain is W.

 

[verty]

V or W could be the real numbers of course, or vector spaces of different dimensions.

 

[kduna]

T is a function that takes a vector $v$ in $V$ to a vector $w$ in $W$. We are very used to the idea of function that take numbers as inputs. For instance, if $f(x) = x^2 + 1$, then $f$ takes $1$ to $2$. We denote this by $f(1) = 2$.

So $T: V \rightarrow W$ means a function that takes v's to w's. I.e. $T(v) = w$.

Now in vector spaces, any old function isn't that useful. We are specifically interested in linear transformations.

$T$ is a linear transformation if the following two properties hold:

$1) \ T(v + v') = T(v) + T(v')$ for all $v, v' \in V$.
$2) \ T(cv) = cT(v)$ for all $v \in V$ and scalars $c$.