Question

Check whether 𝑇 ∶ ℝ2 → ℝ2

, defined by 𝑇 (𝑥, 𝑦) = (−𝑦, 𝑥) is a linear transformation.

Answer 1

A linear transformation (or a linear map) is a function $𝑇 ∶ ℝ^2 → ℝ^2$ that satisfies the following properties:

$$T(a+b)=T(a)+T(b)$$

$$T(\alpha a)=\alpha T(a)$$

for any vectors  $a, b\in \R^2$ and any scalar $\alpha\in \R.$

Let $a=(x_1, y_1), b=(x_2, y_2).$ Then

$$T(a)=(-y_1, x_1)$$

$$T(b)=(-y_2, x_2)$$

$$T(a+b)=(-(y_1+y_2), x_1+x_2)$$

$$=(-y_1, x_1)+(-y_2, x_2)=T(a)+T(b), True$$

$$T(\alpha a)=(-\alpha y_1,\alpha x_1)=\alpha(-y_1, x_1)$$

$$=\alpha T(a), True$$

Therefore a linear transformation $𝑇 ∶ ℝ^2 → ℝ^2$defined by $T(x, y)=(-y,x)$ is a linear transformation.