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# Non-example of a linear transformation

Put content here**Non-example:** A function that fails to be a linear transformation.
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**Example 1 (fails additivity):** The map \(T: \mathbb{R} \to \mathbb{R}\) defined by \(T(x) = x + 1\) is NOT linear because:
\[T(x + y) = x + y + 1 \neq (x + 1) + (y + 1) = T(x) + T(y)\]
Also, \(T(0) = 1 \neq 0\), violating the requirement that linear maps send zero to zero.
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**Example 2 (fails homogeneity):** The map \(T: \mathbb{R}^2 \to \mathbb{R}\) defined by \(T(x,y) = xy\) is NOT linear because:
\[T(2 \cdot (1,1)) = T(2,2) = 4 \neq 2 \cdot T(1,1) = 2 \cdot 1 = 2\]
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**Example 3 (nonlinear function):** The map \(T: \mathbb{R} \to \mathbb{R}\) defined by \(T(x) = x^2\) is NOT linear because \(T(x+y) = (x+y)^2 \neq x^2 + y^2 = T(x) + T(y)\).
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A function is linear only if it preserves both addition and scalar multiplication simultaneously.

# Parents

* Terminology