Definition: The codomain of a linear transformation \(T\) is the vector space \(W\) into which the transformation maps its output vectors.
Notation: If \(T: V \to W\), then \(W\) is the codomain of \(T\).
The codomain specifies the type of output vectors. It may be larger than the actual set of outputs (which is called the range or image).
Example: For \(T: \mathbb{R}^2 \to \mathbb{R}^3\) defined by \(T(x,y) = (x, y, 0)\), the codomain is \(\mathbb{R}^3\), but the range is only the \(xy\)-plane within \(\mathbb{R}^3\).
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