Word Count : 425

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To check whether the function \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \), defined by \( T(x, y) = (-y, x) \), is a linear transformation, we need to verify if it satisfies the two main properties of linearity: 1. Additivity: \( T(u + v) = T(u) + T(v) \) for any vectors \( u, v \in \mathbb{R}^2 \). 2. Homogeneity: \( T(c \cdot u) = c \cdot T(u) \) for any scalar \( c \) and vector \( u \in \mathbb{R}^2 \). Let’s start by checking these properties: ### 1. Additivity Let \( u = (x_1, y_1) \) and \( v = (x_2, y_2) \) be two ____ _____ ____ ___ ________ ________ ______ ________.
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