2.1 Definition and examples of linear transformations
3 min read•august 16, 2024
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
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- preserves vector addition and scalar multiplication between vector spaces
- Two conditions for linearity
- : T(u + v) = T(u) + for all vectors u, v in V
- : 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 of linear transformations
- (null space) of T encompasses all vectors v in V where T(v) = 0
- (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
- of vectors about the origin in R² or R³ (90-degree rotation in xy-plane)
- 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
Key Terms to Review (18)
Additivity: Additivity refers to the property of a function, specifically a linear transformation, where the transformation of the sum of two inputs is equal to the sum of the transformations of each individual input. This means that for any two vectors, when they are added together and then transformed, the result will be the same as transforming each vector separately and then adding those results together. This property is essential for understanding how linear transformations behave and ensures they maintain structure within vector spaces.
Composition of Functions: The composition of functions is the process of combining two functions where the output of one function becomes the input of another. This concept is fundamental in understanding how different transformations interact, particularly in the context of linear transformations, where one linear map can be applied after another, resulting in a new transformation that encapsulates both actions.
Homogeneity: Homogeneity refers to the property of a function, particularly in linear algebra, where if you scale an input by a constant factor, the output is scaled by the same factor. This concept is crucial when discussing linear transformations, as it helps define how these transformations behave under scalar multiplication, ensuring that the structure of the vector space is preserved.
Image: The image of a linear transformation is the set of all output vectors that can be produced by applying the transformation to the input vectors from the domain. It represents the range of the transformation and is crucial for understanding how transformations map elements from one vector space to another. The concept of image is linked to the kernel, as both are essential for characterizing the properties of linear transformations, particularly in terms of their injectivity and surjectivity.
Injective Transformation: An injective transformation is a type of linear transformation that maps distinct elements from one vector space to distinct elements in another, meaning that no two different inputs produce the same output. This characteristic ensures that the transformation preserves the uniqueness of each input, making it a crucial concept in understanding how linear transformations can behave in terms of dimensionality and structure.
Invertibility: Invertibility refers to the property of a linear transformation or a matrix that allows it to be reversed, meaning there exists another transformation or matrix that can undo the action of the original one. This concept is vital in understanding how transformations map inputs to outputs, and when they can be undone, indicating a one-to-one relationship between elements in the domain and codomain.
Isomorphism: Isomorphism is a mathematical concept that describes a structure-preserving mapping between two algebraic structures, such as vector spaces or groups, indicating that they are essentially the same in terms of their properties and operations. This concept highlights how two different systems can be related in a way that preserves the underlying structure, allowing for insights into their behavior and characteristics.
Kernel: The kernel of a linear transformation is the set of all vectors that are mapped to the zero vector. This concept is essential in understanding the behavior of linear transformations, particularly regarding their injectivity and the relationship between different vector spaces. The kernel also plays a crucial role in determining properties like the rank-nullity theorem, which relates the dimensions of the kernel and range.
Linear transformation: A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means if you take any two vectors and apply the transformation, the result will be the same as transforming each vector first and then adding them together. It connects to various concepts, showing how different bases interact, how they can change with respect to matrices, and how they impact the underlying structure of vector spaces.
Matrix Representation: Matrix representation refers to the way a linear transformation is expressed in terms of a matrix that acts on vectors. It allows for the manipulation and analysis of linear transformations in a systematic way by translating the operations into matrix multiplication. This concept is essential in understanding how linear transformations can be simplified, analyzed, and related to properties like eigenvalues and diagonalization.
Rank-Nullity Theorem: The Rank-Nullity Theorem states that for any linear transformation from one vector space to another, the sum of the rank (the dimension of the image) and the nullity (the dimension of the kernel) is equal to the dimension of the domain. This theorem helps illustrate relationships between different aspects of vector spaces and linear transformations, linking concepts like subspaces, linear independence, and matrix representations.
Reflection: Reflection is a type of linear transformation that flips a geometric object over a specific line or plane, creating a mirror image of the original object. It is an important concept in linear algebra as it preserves the structure of the space while altering the position of points within that space. Reflection can be represented mathematically using matrices and is often used in conjunction with other transformations, providing insight into the composition of transformations.
Rotation: Rotation refers to a type of transformation that turns a geometric object around a fixed point, known as the center of rotation, by a certain angle. In the context of linear transformations, rotation is represented by a specific kind of matrix that can alter the position of vectors in a plane while preserving their length. This type of transformation is crucial for understanding how shapes and figures behave under different movements in space.
Scaling: Scaling is a specific type of linear transformation that involves multiplying a vector by a scalar value, which alters the magnitude of the vector while preserving its direction (unless the scalar is negative, which reverses the direction). This concept is foundational in understanding how vectors can be transformed within vector spaces and plays a crucial role in applications such as graphics and physics.
Surjective transformation: A surjective transformation, also known as an onto transformation, is a type of linear transformation where every element in the codomain has at least one pre-image in the domain. This means that the transformation covers the entire codomain, ensuring that there are no 'gaps' or missing values in the output. Understanding surjective transformations is crucial because they help establish relationships between vector spaces and allow for the exploration of dimensionality and function behavior.
T: v → w: The notation 't: v → w' represents a linear transformation 't' that maps vectors from vector space 'v' to vector space 'w'. This mapping preserves the operations of vector addition and scalar multiplication, which are fundamental characteristics of linear transformations. Understanding this mapping is essential as it lays the foundation for examining the kernel and range of transformations, how they can be represented using matrices, and the conditions under which such transformations are invertible.
T(v): In the context of linear transformations, t(v) represents the output of a linear transformation 't' applied to a vector 'v'. This notation is crucial for understanding how a function transforms vectors from one vector space to another, maintaining the properties of vector addition and scalar multiplication. Essentially, t(v) allows us to visualize the effect of a transformation on a vector, providing insight into how linear mappings operate in abstract spaces.
Transformation Theorem: The transformation theorem is a fundamental concept in linear algebra that establishes the conditions under which a linear transformation can be represented by a matrix. This theorem connects linear transformations to matrix representations, ensuring that each linear transformation corresponds uniquely to a matrix once a basis is chosen for the domain and codomain. Understanding this theorem helps in identifying how transformations operate on vector spaces and how matrices can be manipulated to achieve desired transformations.