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.
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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
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
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)
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
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