Fiveable
Fiveable
Fiveable
Fiveable

Linear transformations are functions between vector spaces that preserve addition and scalar multiplication. They're crucial in linear algebra, allowing us to understand how vectors change under different operations. This concept helps us model real-world phenomena and solve complex problems in math and science.

In this part, we'll look at what makes a transformation linear and check out some common examples. We'll learn how to verify if a transformation is linear and explore its geometric meaning. This knowledge will be super helpful for understanding more advanced topics in linear algebra.

Linear Transformations: Definition and Examples

Definition and Properties of Linear Transformations

Top images from around the web for Definition and Properties of Linear Transformations
Top images from around the web for Definition and Properties of Linear Transformations
  • Linear transformation T: V → W preserves vector addition and scalar multiplication between vector spaces
  • Two conditions for linearity
    • Additivity: T(u + v) = T(u) + T(v) for all vectors u, v in V
    • Homogeneity: T(cv) = cT(v) for all vectors v in V and all scalars c
  • Domain V and codomain W constitute vector spaces
  • Finite-dimensional vector spaces allow matrix representation of linear transformations
  • Kernel (null space) of T encompasses all vectors v in V where T(v) = 0
  • Image (range) of T includes all vectors w in W with a corresponding v in V such that T(v) = w

Common Examples of Linear Transformations

  • Rotation of vectors about the origin in R² or R³ (90-degree rotation in xy-plane)
  • Scaling vectors by a constant factor (doubling all vector components)
  • Orthogonal projection onto a subspace (projecting vectors onto x-axis in R²)
  • Differentiation operator D: P_n → P_{n-1} mapping polynomials to derivatives
  • Integration operator mapping functions to indefinite integrals
  • Matrix multiplication by fixed matrix A defining transformation from R^n to R^m
  • Zero transformation mapping every vector to zero vector

Verifying Linear Transformations

Methods for Verifying Linearity

  • Check additivity and homogeneity properties for arbitrary vectors and scalars
  • Additivity verification T(u + v) = T(u) + T(v) for all vectors u and v in domain
  • Homogeneity verification T(cv) = cT(v) for all vectors v in domain and scalars c
  • Utilize counterexamples to disprove linearity by finding vectors or scalars violating properties
  • Apply algebraic manipulation to verify linearity for formula-defined transformations
  • Verify linearity on basis for finite-dimensional vector spaces to prove linearity for entire space

Examples of Linearity Verification

  • Rotation transformation in R²: T(x, y) = (-y, x)
    • Verify additivity: T((x₁, y₁) + (x₂, y₂)) = T(x₁ + x₂, y₁ + y₂) = (-(y₁ + y₂), x₁ + x₂) = (-y₁, x₁) + (-y₂, x₂) = T(x₁, y₁) + T(x₂, y₂)
    • Verify homogeneity: T(c(x, y)) = T(cx, cy) = (-cy, cx) = c(-y, x) = cT(x, y)
  • Non-linear transformation example: T(x) = x² + 1
    • Counterexample for additivity: T(2 + 3) ≠ T(2) + T(3), as 5² + 1 ≠ (2² + 1) + (3² + 1)

Geometric Interpretation of Linear Transformations

Preservation Properties of Linear Transformations

  • Origin preservation maps zero vector to zero vector
  • Lines transform to lines or points if passing through origin
  • Parallel lines maintain parallelism after transformation
  • Domain grid transforms into new codomain grid with straight lines remaining straight
  • Point collinearity preserved under linear transformations
  • Ratio of parallel line segment lengths maintained

Visualizing Linear Transformations

  • R² transformations visualized through effect on unit square or basis vectors
    • Shear transformation: T(x, y) = (x + y, y) stretches unit square into parallelogram
  • R³ transformations understood by effect on unit cube or basis vectors
    • Rotation about z-axis: T(x, y, z) = (x cos θ - y sin θ, x sin θ + y cos θ, z) rotates unit cube
  • Linear transformations maintain grid structure (squares to parallelograms, cubes to parallelepipeds)
  • Composition of linear transformations visualized as sequential application of individual transformations
© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Glossary
Glossary