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Definition LT. Linear Transformation. A linear transformation, T : U â V , is a function that carries elements of the vector space U (called the domain) to the vector space V (called the codomain), and which has two additional properties.
Definition IVS Isomorphic Vector Spaces. Two vector spaces $U$ and $V$ are isomorphic if there exists an invertible linear transformation $T$ with domain $U$ and codomain $V$, $\ltdefn{T}{U}{V}$. In this case, we write $U\isomorphic V$, and the linear transformation $T$ is known as an isomorphism between $U$ and $V$.A few comments on this definition.heorem ICLT Inverse of a Composition of Linear Transformations. Suppose that $\ltdefn{T}{U}{V}$ and $\ltdefn{S}{V}{W}$ are invertible linear transformations. Then $\compose{S}{T}$ is invertible and $\ltinverse{\left(\compose{S}{T}\right)}=\compose{\ltinverse{T}}{\ltinverse{S}}$.