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Non-example of a linear transformation

Created over 8 years ago, updated 10 days ago

Non-example: A function that fails to be a linear transformation.

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.

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]

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)).

A function is linear only if it preserves both addition and scalar multiplication simultaneously.