A linear map is a function that takes vectors, transforms them into new vectors, and makes sure that adding and scaling vectors still works the same way before and after the transformation. A linear map is just a rule that takes a vector and turns it into another vector without bending or breaking its structure. It respects addition and scaling. Think of it like applying a fixed transformation that preserves lines and the origin.
Definition
A linear map (or linear transformation) is a function \(T: V \to W\) between two [[Vector Space]] such that for all vectors \(\mathbf{u}, \mathbf{v} \in V\) and all scalars \(c \in \mathbb{R}\) (or any field), it satisfies:
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Additivity: Transformation on addition is addition of transformations: \(T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})\)
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Homogeneity (scalar multiplication): Transformation on a vector scalar product is same as scalar multiplied to a transformed vector. \(T(c\mathbf{v}) = cT(\mathbf{v})\)
These two together mean: $$ T(a\mathbf{u} + b\mathbf{v}) = aT(\mathbf{u}) + bT(\mathbf{v}) $$
Key Properties
- The origin maps to origin: \(T(\mathbf{0}) = \mathbf{0}\)
- Linear maps can be represented as matrices when vector spaces are finite-dimensional.
- If \(T: \mathbb{R}^n \to \mathbb{R}^m\), then \(T(\mathbf{x}) = A\mathbf{x}\) for some \(m \times n\) matrix \(A\).
Examples
- Scaling: \(T(x) = 2x\) → doubles every input → linear.
- Rotation in the plane → linear.
- \(T(x) = x + 1\) → not linear because it doesn’t preserve the origin.
Geometric Intuition
A linear map: * Preserves straight lines. * Preserves ratios along lines. * Doesn’t twist, curve, or displace the origin.
Why It Matters
Linear maps are the building blocks of all transformations in linear algebra. Every matrix is a linear map. They're fundamental in:
- Solving systems of equations.
- Understanding projections, rotations, shears.
- Machine learning (every neural net layer is mostly a linear map + nonlinearity).