Question

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)

Answer 1

Answer on Question #55826 – Math – Linear Algebra

Let $V$ be a vector space over a field $F$ and function

$I\colon V\to V$ be the identity transformation. Find range of $I\colon R(I)$

Solution

The range of operator $A\colon V \to V$ is a 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: $V$

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