Question

1 Let V be a vector space over a field F and function
I:V→V
be the identity transformation. Find range of I R(I)
{0}
{1}
{0,1}
none of the above

Answer 1

The range of operator $A: V \to V$ is set of values it reaches:

$$R(A) = \{v \in V \mid \exists w \in V: v = A(w)\}$$

The identity operator is defined as

$$I(v) = v, \forall v \in V$$

Thus the range of identity operator $I$ is:

$$R(I) = \{v \in V \mid \exists w \in V: v = I(w)\} = \{v \in V \mid \exists w \in V: v = w\} = V$$

ANSWER: none of the above