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Linear Algebra: Deriving Rotation and Reflection Matrices from Standard Basis Vectors in R2

In linear algebra, every linear transformation from $\mathbb{R}^n$ to a vector space is uniquely represented by an $n \times n$ matrix where each column corresponds exactly to the image of the respective standard basis vector under that transformation. This principle establishes that knowing the action of a function on only $n$ specific vectors (the standard basis) constitutes sufficient information to completely characterize its behavior for all possible input vectors within the domain.