Question #342820

Check whether 𝑇 ∶ ℝ2 → ℝ2

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

Expert's answer

A linear transformation (or a linear map) is a function "\ud835\udc47 \u2236 \u211d^2 \u2192 \u211d^2" that satisfies the following properties:

"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(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 "\ud835\udc47 \u2236 \u211d^2 \u2192 \u211d^2"defined by "T(x, y)=(-y,x)" is a linear transformation.

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