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How is a Map Linear?

Published in Linear Algebra 3 mins read

A map, often referred to as a transformation or function, is considered linear if it adheres to specific rules regarding how it handles vector addition and scalar multiplication. These rules ensure that the structure of the vector space is preserved through the map.

Understanding Linear Maps

A linear map, denoted as T : V → W, where V and W are vector spaces, must satisfy two crucial conditions according to the provided reference:

  1. Additivity: The map of the sum of two vectors is equal to the sum of the maps of the individual vectors.

    • Mathematically: T(X + Y) = T(X) + T(Y) for any vectors X and Y in V.
    • This means that if you add two vectors first, and then apply the map, it gives the same result as applying the map to each vector separately before adding the results.

  2. Homogeneity (Scalar Multiplication): The map of a scalar multiple of a vector is equal to the scalar multiple of the map of the vector.

    • Mathematically: T(λX) = λT(X) for any vector X in V and any scalar λ from field F.
    • This property ensures that scaling a vector before the map is the same as scaling the mapped vector.

Examples

Let's clarify with examples:

  • Linear Map: Consider the map T(x,y) = (2x, y + x).
    • Additivity Check:
      • Let X = (x1, y1) and Y = (x2, y2).
      • T(X+Y) = T(x1+x2, y1+y2) = (2(x1+x2), (y1+y2)+(x1+x2)) = (2x1+2x2, y1+y2+x1+x2).
      • T(X) + T(Y) = (2x1, y1+x1) + (2x2, y2+x2) = (2x1+2x2, y1+x1+y2+x2).
      • Therefore, T(X+Y) = T(X) + T(Y), and this additivity property holds.
    • Homogeneity Check:
      • T(λX) = T(λx1, λy1) = (2λx1, λy1 + λx1) = (λ(2x1), λ(y1 + x1)) = λ(2x1, y1+x1) = λT(X).
      • Therefore, T(λX) = λT(X), and this homogeneity property holds.
    • Since both conditions are satisfied, T is a linear map.
  • Non-Linear Map: Consider the map T(x, y) = (x^2, y).
    • Additivity Check: Let X = (1,1) and Y = (2,2)
      • T(X+Y) = T(3,3) = (9, 3)
      • T(X) + T(Y) = (1, 1) + (4, 2) = (5, 3)
      • T(X+Y) != T(X) + T(Y) so additivity does not hold
      • Since at least one condition is not met, this is not a linear map

Why Linearity is Important

Linear maps are fundamental in many areas of mathematics, physics, and computer science because they preserve essential structures:

  • They allow for easy manipulation and analysis using matrix operations.
  • They are often used to model physical systems that exhibit linear behavior.
  • They serve as building blocks for more complex transformations and algorithms.

In Summary

A map is linear if it preserves vector addition and scalar multiplication. Specifically, the reference states that T(X + Y) = T(X) + T(Y) and T(λX) = λT(X) for a map to be considered linear.

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