Glossary

2.4 Linear transformations and basis functions

3 min read•july 18, 2024

Linear transformations are the backbone of signal processing, allowing us to manipulate and analyze complex data. They preserve key relationships between vectors, making them invaluable for understanding how signals change and interact in various systems.

Basis functions provide a framework for representing signals in different domains. By changing the basis, we can simplify complex signals, extract important features, and apply transformations that reveal hidden patterns in the data.

Linear Transformations

Properties of linear transformations

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  • Preserve vector addition enables combining transformed vectors in the same way as the original vectors
  • Preserve scalar multiplication allows scaling transformed vectors by the same factor as the original vectors
  • Map the zero vector to the zero vector maintains the origin of the
  • Preserve linear combinations ensures that any linear combination of vectors is transformed into the corresponding linear combination of their transformed counterparts

Matrix representation of transformations

  • AA contains the images of the standard basis vectors e1,e2,…,ene_1, e_2, \ldots, e_n as its columns
  • Matrix multiplication AxAx computes the of vector xx by mapping each component to its corresponding transformed value
  • Composition of linear transformations corresponds to matrix multiplication of their respective transformation matrices (T2∘T1↔A2A1T_2 \circ T_1 \leftrightarrow A_2A_1)
  • Inverse of a linear transformation, if it exists, is represented by the inverse of its transformation matrix (T−1↔A−1T^{-1} \leftrightarrow A^{-1})

Basis Functions

Role of basis functions

  • Provide a coordinate system for representing vectors in a vector space or signals in a function space
  • Enable unique representation of any vector or signal as a linear combination of the basis functions
  • Allow for compact and efficient representation of signals by capturing their essential features
  • Facilitate analysis, processing, and transformation of signals in different domains (time↔frequency\text{time} \leftrightarrow \text{frequency})

Basis changes for signals

  • AA contains the new basis vectors expressed in terms of the original basis
  • Coefficients in the new basis cnewc_{\text{new}} are obtained by multiplying the original coefficients coldc_{\text{old}} by the change of basis matrix: cnew=Acoldc_{\text{new}} = Ac_{\text{old}}
  • A−1A^{-1} maps the coefficients back to the original basis: cold=A−1cnewc_{\text{old}} = A^{-1}c_{\text{new}}
  • Basis changes can be used to simplify signal representation, extract features, or apply transformations

Common basis functions in processing

  • represents signals as a sum of complex exponentials with different frequencies
    • Enables frequency-domain analysis and filtering (low-pass\text{low-pass}, high-pass\text{high-pass}, band-pass\text{band-pass})
    • Used in applications such as audio processing, telecommunications, and radar
  • represents signals at different scales and positions using scaled and shifted versions of a mother wavelet
    • Captures both frequency and time information, providing a multi-resolution analysis
    • Used in applications such as image compression (JPEG 2000\text{JPEG 2000}), denoising, and feature extraction
  • represents signals as a sum of powers of a variable (11, xx, x2x^2, …\ldots)
    • Used in curve fitting, interpolation, and approximation problems
  • Legendre and Hermite bases are orthogonal polynomial bases used in solving differential equations and quantum mechanics, respectively

Key Terms to Review (33)

Affine transformation: An affine transformation is a mathematical operation that preserves points, straight lines, and planes. It is defined by a linear mapping followed by a translation, allowing for operations such as scaling, rotation, translation, and shearing. This transformation maintains the collinearity of points and the ratios of distances between points, making it crucial in applications such as computer graphics and image processing.
Affine Transformation: An affine transformation is a mathematical operation that combines linear transformations and translations, preserving points, straight lines, and planes. This type of transformation allows for scaling, rotation, translation, and shearing of objects within a coordinate space while maintaining the parallelism of lines and ratios of distances. It plays a crucial role in understanding how changes in coordinates can affect the representation of data in various applications.
Change of Basis Matrix: A change of basis matrix is a transformation matrix that converts coordinates of vectors from one basis to another in a vector space. It allows for the representation of the same vector in different coordinate systems, thereby facilitating computations and interpretations in various contexts. This concept is closely linked to linear transformations, as it enables the mapping of vector representations while preserving their inherent properties.
Change of Basis Theorem: The Change of Basis Theorem states that any vector in a vector space can be represented in terms of different bases, allowing for a transformation between these bases. This theorem is essential because it helps simplify problems by allowing calculations to be done in a more convenient basis, making it easier to analyze linear transformations and understand their effects on various functions.
Continuity: Continuity refers to the property of a function where small changes in the input result in small changes in the output. This concept is crucial when discussing linear transformations and basis functions, as it ensures that the output remains predictable and stable as inputs vary. Understanding continuity allows for smooth transitions and mappings between vector spaces, which is essential for analyzing and solving problems in bioengineering and signal processing.
Dimension: Dimension refers to the number of independent directions or parameters in a mathematical space, which is essential for understanding the structure of that space. It connects to concepts like linear transformations and basis functions by indicating how many vectors are needed to represent a vector space completely. This understanding is fundamental for analyzing how different transformations can map vectors from one space to another and how these mappings can be characterized in terms of their dimensionality.
Eigenvalue decomposition: Eigenvalue decomposition is a mathematical technique that expresses a square matrix as a product of its eigenvalues and eigenvectors. This process provides insights into the properties of linear transformations and helps in analyzing systems by breaking down complex operations into simpler components.
Fourier Basis: The Fourier basis refers to a set of orthogonal functions used to represent periodic signals as sums of sine and cosine functions. This concept is essential in breaking down complex signals into simpler components, facilitating the analysis and manipulation of signals in various fields, including engineering and physics.
Hermite Basis: A Hermite basis refers to a set of orthogonal basis functions derived from Hermite polynomials, which are used extensively in approximation theory and numerical analysis. These basis functions allow for the representation of functions in a way that preserves certain properties, such as smoothness and continuity, making them valuable in applications like signal processing and control systems.
Hilbert Space: A Hilbert space is a complete inner product space that serves as a fundamental concept in functional analysis, providing a geometric framework for the study of infinite-dimensional vector spaces. It extends the notion of Euclidean space, allowing for the rigorous treatment of concepts like convergence and orthogonality, making it essential in various applications including quantum mechanics and signal processing.
Identity matrix: An identity matrix is a special kind of square matrix that serves as the multiplicative identity in matrix multiplication. When any matrix is multiplied by an identity matrix of compatible dimensions, the original matrix remains unchanged. This unique property makes the identity matrix crucial in linear transformations, especially when discussing transformations related to basis functions and their effects on vector spaces.
Image Processing: Image processing is a technique used to enhance, analyze, and manipulate images through various algorithms and transformations. It enables the extraction of useful information from images and can be applied in multiple fields such as medical imaging, computer vision, and remote sensing. Understanding how to apply linear transformations and basis functions is crucial in image processing as they form the mathematical foundation for many of the techniques used to modify and interpret images.
Image Reconstruction: Image reconstruction refers to the process of creating a visual representation from raw data, often captured through various imaging modalities. This technique is vital in fields like medical imaging, where it transforms raw signals into clear images that clinicians can analyze. The quality of the reconstructed image depends heavily on the algorithms used and the data acquisition methods employed, influencing how well details can be visualized and interpreted.
Inverse change of basis matrix: An inverse change of basis matrix is a matrix that transforms coordinate representations of vectors from one basis to another, effectively reversing the transformation applied by the original change of basis matrix. This concept is crucial when working with linear transformations, as it allows for the conversion between different vector spaces while maintaining the relationships and properties of the original vectors.
Invertibility: Invertibility refers to the property of a linear transformation where there exists an inverse transformation that can reverse the effect of the original transformation. In the context of linear systems, this means that for every output vector produced by the transformation, there is a unique input vector that can be retrieved, making it essential for solving equations and ensuring a one-to-one relationship between inputs and outputs.
Jordan Canonical Form: Jordan Canonical Form is a special type of matrix representation that simplifies the study of linear transformations, particularly for matrices that cannot be diagonalized. It organizes a matrix into blocks, called Jordan blocks, that correspond to its eigenvalues and their geometric and algebraic multiplicities, making it easier to analyze the behavior of linear transformations and understand their effects on basis functions.
L2 space: l2 space, also known as the space of square-summable sequences, is a mathematical space consisting of infinite-dimensional vectors where the sum of the squares of their components is finite. This concept is fundamental in functional analysis and is closely related to linear transformations and basis functions, as it provides a framework for understanding how these transformations can be applied to vectors in a structured manner. l2 space serves as a complete inner product space, enabling the exploration of various properties like convergence and continuity of linear operators acting on these sequences.
Least Squares Approximation: Least squares approximation is a mathematical method used to find the best-fitting curve or line for a set of data points by minimizing the sum of the squares of the vertical distances (residuals) between the data points and the curve. This method is fundamental in linear transformations, as it helps in determining coefficients of basis functions that best represent data in various applications, such as signal processing and system modeling.
Legendre Basis: The Legendre basis is a set of orthogonal polynomials that arise in the context of functional analysis and approximation theory. These polynomials are defined on the interval [-1, 1] and are particularly useful for solving problems involving linear transformations and expansions in series. The properties of these polynomials, such as their orthogonality and completeness, make them ideal for representing functions in various applications, including numerical methods and signal processing.
Linear Mapping: Linear mapping refers to a mathematical function that transforms vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This concept is crucial in understanding how different functions can be represented and manipulated in mathematical spaces, particularly when analyzing systems that exhibit linear behavior.
Linear Transformation: A linear transformation is a mathematical function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. This means that if you take two vectors and add them together or multiply a vector by a scalar, the transformation will yield consistent results, maintaining the structure of the vector space. Linear transformations are fundamental in connecting various concepts, particularly in understanding how functions can be represented in terms of matrix operations.
Linearity: Linearity refers to the property of a system or transformation where the output is directly proportional to the input, following the principles of superposition. This means that if you combine inputs, the output will be a combination of the outputs produced by each input separately. Linearity is crucial in many areas of signal processing and systems analysis, as it allows for simplified analysis and predictable behavior of systems under various conditions.
Matrix Diagonalization: Matrix diagonalization is the process of transforming a matrix into a diagonal form, where all its non-diagonal elements are zero, making it easier to work with in various applications, particularly in solving systems of equations and performing linear transformations. This technique is closely connected to finding the eigenvalues and eigenvectors of the matrix, which provide insights into its geometric properties and allow for simplifications in calculations.
Matrix representation: Matrix representation is a way to express linear transformations and systems of equations using matrices, allowing for a structured and efficient approach to manipulating and analyzing data. This concept connects closely to how functions transform inputs into outputs while maintaining relationships between different dimensions in vector spaces. By using matrices, we can simplify complex operations and leverage computational tools to solve problems in various fields.
Null Space: The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector. This concept is essential for understanding linear transformations, as it describes the input vectors that do not produce any output, highlighting the limitations and behavior of the transformation. The null space is also important in determining the rank and dimension of a matrix, connecting to the concept of basis functions, which can be used to represent these vectors efficiently.
Orthonormal Basis: An orthonormal basis is a set of vectors that are both orthogonal and normalized, meaning they are perpendicular to each other and have a unit length. This concept is essential in linear algebra, particularly in understanding how linear transformations can be simplified and analyzed. When a space is described using an orthonormal basis, any vector within that space can be represented as a unique linear combination of these basis vectors, which simplifies computations and enhances numerical stability in various applications.
Polynomial Basis: A polynomial basis is a set of polynomial functions that span a vector space, allowing any polynomial in that space to be expressed as a linear combination of the basis polynomials. This concept is crucial in understanding linear transformations and the representation of functions within certain dimensions, enabling efficient computation and analysis in various applications, especially in signal processing and bioengineering.
Rank: In linear algebra, the rank of a matrix refers to the dimension of the vector space generated by its rows or columns. This concept is fundamental as it helps determine the effectiveness of linear transformations and the relationships between basis functions, ultimately influencing the solutions to linear equations and the structure of the associated vector spaces.
Rank-Nullity Theorem: The rank-nullity theorem states that for any linear transformation from one vector space to another, the dimension of the domain is equal to the sum of the rank and the nullity of the transformation. This relationship highlights the balance between the dimensions of images and kernels in the context of linear mappings, illustrating how much information is preserved or lost during transformation.
Singular Value Decomposition: Singular Value Decomposition (SVD) is a mathematical technique used to factor a matrix into three distinct matrices, revealing essential properties and features of the original matrix. This decomposition can simplify complex operations in signal processing, such as data compression and noise reduction. By breaking down a matrix into its singular values and vectors, SVD enables effective representation and manipulation of data in various applications, particularly when working with high-dimensional spaces or transforming datasets.
Transformation matrix: A transformation matrix is a mathematical construct that represents a linear transformation from one vector space to another. It acts on vectors to change their position, orientation, or size in a systematic way, allowing for operations like rotation, scaling, and translation. This matrix can be applied to coordinate systems and basis functions to manipulate geometric and algebraic representations of data in various fields, such as computer graphics and engineering.
Vector Space: A vector space is a collection of vectors that can be added together and multiplied by scalars, satisfying certain properties such as closure, associativity, and distributivity. This mathematical structure is fundamental in various fields, as it provides a framework for analyzing linear combinations, transformations, and function spaces. In the context of linear transformations and basis functions, understanding vector spaces helps in visualizing and manipulating the relationships between different sets of vectors and their respective transformations.
Wavelet Basis: A wavelet basis is a set of functions generated from a single prototype function called the mother wavelet, which allows for representing signals and data at various scales and resolutions. This concept is crucial in transforming signals into different forms, making it easier to analyze their features and properties through linear transformations. Wavelet bases provide flexibility in signal representation, especially for non-stationary signals, allowing them to capture both time and frequency information effectively.
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