gen-state-a-linear-map-precisely-and-explain-why-it-matters-in-2-175f

State a linear map precisely and explain why it matters in 2-3 sentences.

## Definition A linear map is a function $T: V \to W$ between vector spaces that preserves vector addition and scalar multiplication: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{v}) = cT(\mathbf{v})$ for all vectors $\mathbf{u}, \mathbf{v} \in V$ and scalars $c$. ## Why It Matters Linear maps are fundamental because they preserve the structure of vector spaces, making them predictable and computationally tractable—once you know where basis vectors go, you know where everything goes. They appear everywhere in mathematics and applications: rotations, projections, derivatives, quantum mechanics, machine learning, and any system where outputs scale proportionally with inputs.