Question

Let U and V be vector spaces over a field F. Let
T:U→V
be a linear transformation, then the set
[T(x)]x∈U
is called the
transformation
space
kernel of T
range of T

Answer 1

The range of operator $T: U \to V$ is defined as:

$$R(A) = \{v \in V: \exists u \in U: v = A(u)\}$$

Rewriting this definition we got:

$$\{v \in V: \exists u \in U: v = A(u)\} = \{T(u) \mid u \in U\} = T(U)$$

Using this we conclude that $T(U)$ is the range of linear transformation $T$.

ANSWER: range of T