Question

Determine which of the following transformations l: r2 → r2 is linear?

1) l(x) = x²
2) l(x) = 2x
3) l(x) = x + 1
4) l(x) = 3x²

Answer 1

Only the transformation ℒ(x) = 2x is linear, as it meets the required properties of additivity and homogeneity (scalar multiplication) for linear transformations.

Step-by-step explanation:

To determine which of the given transformations ℒ: ℝ² → ℝ² is linear, we must check if they satisfy two main properties of linear transformations: additivity and homogeneity (scalar multiplication). Let's apply these conditions to each of the given transformations.

1. ℒ(x) = x² does not satisfy the additivity property because ℒ(x+y) ≠ ℒ(x) + ℒ(y), thus it is not linear.
2. ℒ(x) = 2x satisfies both the additivity and homogeneity properties, hence it is a linear transformation.
3. ℒ(x) = x + 1 does not satisfy the additivity property since ℒ(0) does not equal 0, and therefore it is not linear.
4. ℒ(x) = 3x² does not satisfy the additivity property as well, similar to the first transformation, making it non-linear.

Thus, the only transformation that is linear is ℒ(x) = 2x.