The range is a subspace
If , then range is a subspace of .
Proof. Suppose . Then because linear maps take 0 to 0, which implies that .
If , then there exist such that and . Thus
Hence, . Thus, range is closed under addition.
If and , then there exists such that . Thus
Hence, . Thus is closed under scalar multiplication.
We have shown that contains , is closed under addition, and is closed under scalar multiplication. Thus, is a subspace of .