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Let $$V$$ and $$W$$ be real vector spaces (their dimensions can be different), and let $$T$$ be a function with domain $$V$$ and range in $$W$$ (written $$T: V \rightarrow W)$$. We say $$T$$ is a linear transformation if
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Let $$V$$ and $$W$$ be real vector spaces (their dimensions can be different), and let $$T$$ be a function with domain $$V$$ and range in $$W$$ (written $$T: V \rightarrow W)$$. We say $$T$$ is a linear transformation if

(a) For all $$\mathbf{x}, \mathbf{y} \in V, T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y})(T$$ is additive $$)$$.

(b) For all $$\mathbf{x} \in V, r \in \mathbb{R}, T(r \mathbf{x})=r T(\mathbf{x})$$ ( $$T$$ is homogeneous).
If $$V$$ and $$W$$ are complex vector spaces, the definition is the same except in (b), $$r \in \mathbb{C}$$. If $$V=W$$, then $$T$$ can be called a linear operator.
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