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.
**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.