Definition: The domain of a linear transformation \(T\) is the vector space \(V\) from which the transformation takes its input vectors.
Notation: If \(T: V \to W\), then \(V\) is the domain of \(T\).
Every vector in the domain must be mapped to exactly one vector in the codomain. The domain is the entire source vector space, not just a subset.
Example: For the transformation \(T: \mathbb{R}^3 \to \mathbb{R}^2\) defined by \(T(x,y,z) = (x+y, z)\), the domain is \(\mathbb{R}^3\). Any vector in \(\mathbb{R}^3\) can serve as input to \(T\).
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