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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.
up vote 1 down vote accepted. See the kernel of the linear transformation. If it contains only the zero vector, then the linear transformation is one to one and hence an isomorphism (as the two spaces in question are of the same finite dimension).Clearly, $ker T=\{(x,y):T(x,y)=(0,0)\}=\{(x,y):(x-y,x+y)=(0,0)\}=\{(x,y):x-y=0,x+y=0\}$.y the definition the linear isomorphism is a linear bijective map. We should show that for any point $(x,y)\in \mathbb R^2$ there exists unique point $(x',y')\in \mathbb R^2$ such that $f(x',y') = (x,y)$ (this point is called pre-image of $(x,y)$). Note that these any + unique are properties of a bijection.