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A vector is a mathematical object that has both magnitude and direction, and is used to represent quantities such as force, velocity, and displacement in physics and engineering. Vectors are fundamental in linear algebra and are often represented as an ordered list of numbers, which can be manipulated using operations like addition, subtraction, and scalar multiplication.
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A linear combination involves summing multiple vectors, each multiplied by a scalar coefficient, to form a new vector in the same vector space. This concept is fundamental in linear algebra and is used in various applications such as solving linear equations, transformations, and understanding vector spaces and their spans.
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In finance and investing, 'basis' refers to the difference between the spot price of an asset and its corresponding futures price. It is a critical metric for traders and investors as it helps in assessing the cost of carry and potential arbitrage opportunities in futures markets.
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In mathematics and physics, a dimension is an independent direction in space, often used to describe the structure and behavior of objects and phenomena in the universe. Dimensions can be spatial, temporal, or abstract, and they play a crucial role in understanding the geometry, topology, and dynamics of different systems.
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A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication, mapping lines to lines or points through the origin. These transformations can be represented by matrices, making them fundamental in solving systems of linear equations and understanding geometric transformations in higher dimensions.
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A subspace is a subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication. It must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication to qualify as a subspace.
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Span in linear algebra refers to the set of all possible linear combinations of a given set of vectors, essentially describing the space that these vectors can cover. Understanding the span is crucial for determining vector spaces, subspaces, and for solving systems of linear equations.
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Scalar multiplication involves multiplying a vector by a scalar, resulting in a new vector that is scaled in magnitude but retains the same direction unless the scalar is negative, which reverses the direction. This operation is fundamental in linear algebra and is used to scale vectors in various applications, such as physics and computer graphics.
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Vector addition is a fundamental operation in vector algebra that combines two or more vectors to produce a resultant vector. It follows the triangle or parallelogram law, ensuring that the resultant vector maintains both magnitude and direction based on the components of the original vectors.
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The zero vector is a vector in a vector space that has all of its components equal to zero, serving as the additive identity. It plays a crucial role in linear algebra as it is the only vector that, when added to any other vector, leaves the other vector unchanged.
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A 'norm' is a standard or rule that is socially enforced, guiding behavior within a society or group. Norms can be explicit, such as laws, or implicit, like cultural customs, and they play a crucial role in maintaining social order and cohesion.
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The inner product is a fundamental operation in linear algebra that generalizes the dot product to abstract vector spaces, providing a way to define angles and lengths. It is essential for understanding orthogonality, projections, and the structure of Hilbert spaces, with applications across mathematics and physics.
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Orthogonality is a fundamental concept in mathematics and engineering that describes the relationship between two vectors being perpendicular, meaning their dot product is zero. This concept extends beyond geometry to functions, signals, and data analysis, where orthogonality implies independence and non-interference among components.
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An eigenvector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied. Eigenvectors are fundamental in simplifying matrix operations, especially in diagonalization, stability analysis, and principal component analysis.
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An eigenvalue is a scalar that indicates how much the eigenvector is stretched or compressed during a linear transformation represented by a matrix. In essence, eigenvalues reveal important properties of the matrix, such as stability and resonance in physical systems, and are fundamental in simplifying complex matrix operations like diagonalization.
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Linear algebra is a branch of mathematics that deals with vector spaces and linear mappings between these spaces, focusing on the study of lines, planes, and subspaces. It is fundamental in various scientific fields, providing tools for solving systems of linear equations, performing transformations, and analyzing vector spaces and matrices.
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Matrix notation is a compact and efficient way to represent and manipulate arrays of numbers, which is essential in various fields such as mathematics, physics, computer science, and engineering. It allows for the concise expression of linear equations and transformations, facilitating operations like addition, multiplication, and inversion of matrices.
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The weighted inner product is a generalization of the standard inner product, where each dimension of the vectors involved is scaled by a corresponding weight. This allows for more flexible similarity measures in applications such as machine learning, where different features may have varying levels of importance.
Eigenvalues and eigenfunctions are fundamental in understanding how linear transformations affect vector spaces, particularly in solving differential equations and quantum mechanics. They reveal intrinsic properties of operators by identifying invariant directions and scaling factors, simplifying complex systems into more manageable forms.
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Matrix transpose involves flipping a matrix over its diagonal, effectively swapping the row and column indices of each element. It is a fundamental operation in linear algebra with applications in various fields such as computer graphics, data analysis, and machine learning.
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The Gram-Schmidt process is an algorithm for orthogonalizing a set of vectors in an inner product space, often used to convert a basis into an orthonormal basis. It is fundamental in numerical linear algebra, facilitating processes like QR decomposition and improving the stability of computations involving vectors.
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An orthogonal region is a geometric area in a multi-dimensional space where all axes are perpendicular to each other, allowing for simplified mathematical operations and analysis. This concept is crucial in fields like linear algebra and computer graphics, where it aids in transformations and optimizations by providing a clear, non-overlapping structure.
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Coordinate systems provide a framework for defining the position of points in space, using a set of numbers called coordinates. They are essential in mathematics, physics, and engineering for describing spatial relationships and transformations between different reference frames.
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High-dimensional spaces refer to mathematical spaces with a large number of dimensions, often used in fields like machine learning and data analysis to represent complex data structures. These spaces pose unique challenges such as the 'curse of dimensionality,' which can lead to increased computational complexity and data sparsity, affecting the performance of algorithms.
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Multilinear maps are mathematical functions that take multiple vector inputs and return a single scalar, preserving linearity in each argument when the others are held constant. They are fundamental in tensor algebra and have applications in cryptography, particularly in constructing advanced cryptographic protocols like Multilinear maps-based cryptosystems.
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Tensor products are a mathematical construction that allow for the combination of vector spaces, facilitating the representation of multilinear relationships. They are crucial in fields such as quantum mechanics, where they enable the description of composite systems by combining state spaces of individual components.
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An affine combination is a linear combination of vectors where the coefficients sum to one, allowing for the representation of points in affine spaces. It is a fundamental concept in geometry and linear algebra, often used in computer graphics, optimization, and data interpolation.
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Geometric representation is a mathematical approach used to visualize and understand abstract concepts by mapping them onto geometric objects, allowing for intuitive insights and problem-solving. This method is widely used across various fields such as computer graphics, data visualization, and theoretical physics to simplify complex systems and facilitate communication of ideas.
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The transpose of a matrix is obtained by swapping its rows with columns, effectively flipping the matrix over its diagonal. This operation is fundamental in linear algebra, playing a crucial role in matrix operations, vector spaces, and applications like solving systems of equations and computer graphics.
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Mathematical structures are abstract frameworks that encapsulate sets along with operations and relations, providing a foundation for various branches of mathematics. They enable the formalization of mathematical theories and facilitate the study of properties and relationships within these theories.
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