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To be a vector space isomorphism, a map $F$ must preserve vector addition and scalar multiplication. Specifically, $F(a+b) = F(a) + F(b)$ and $F(ka) = kF(a)$ for any vectors $a, b$ and scalar $k$.

Conditions: $a, b \in V$ are arbitrary vectors.; $k$ is a scalar in the underlying field.

To be a vector space isomorphism, a map $F$ must preserve vector addition and scalar multiplication. Specifically, $F(a+b) = F(a) + F(b)$ and $F(ka) = kF(a)$ for any vectors $a, b$ and scalar $k$.

Conditions: $a, b \in V$ are arbitrary vectors.; $k$ is a scalar in the underlying field.

A linear isometry is a map $F: V \to W$ between two inner product spaces that is both a vector space isomorphism and preserves inner products. This means it maintains the geometric structure defined by the inner product.

Conditions: $V$ and $W$ are inner product spaces.; $F$ is a vector space isomorphism.; $F$ preserves inner products for all $a, b \in V$.

A linear isometry is a map $F: V \to W$ between two inner product spaces that is both a vector space isomorphism and preserves inner products. This means it maintains the geometric structure defined by the inner product.

Conditions: $V$ and $W$ are inner product spaces.; $F$ is a vector space isomorphism.; $F$ preserves inner products for all $a, b \in V$.