Range of linear transformation is subspace of codomain

Dependencies:

1. Linear transformation
2. Condition for being a subspace

Let $T: U \mapsto V$ be a linear transformation. Then the range of $T$ (denoted as $T(U)$) is a subspace of $V$.

Proof

\[ T(x), T(y) \in T(U) \implies T(x) + T(y) = T(x+y) \in T(U) \] \[ T(x) \in T(U) \implies cT(x) = T(cx) \in T(U) \]

Therefore, $T(U)$ is a subspace of $V$.

Dependency for:

1. Basis of range of linear transformation

Info:

• Depth: 7
• Number of transitive dependencies: 13

Transitive dependencies:

1. Group
2. Identity of a group is unique
3. Subgroup
4. Inverse of a group element is unique
5. Conditions for a subset to be a subgroup
6. Condition for a subset to be a subgroup
7. Ring
8. Field
9. Vector Space
10. Zeros in vector space
11. Negation in vector space
12. Condition for being a subspace
13. Linear transformation