Questions and Answers
What is algebraic multiplicity?
What defines a basic variable in a linear system?
What is a basis?
An indexed set B = {v1,...,vP} in V that is linearly independent and spans H.
What is the best approximation?
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What is the characteristic equation?
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What is the codomain of a transformation T?
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What is the column space?
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A consistent linear system is one with no solutions.
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What does it mean for a matrix to be diagonalizable?
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Define a diagonal matrix.
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What is the dimension of a subspace S?
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What is the domain of a transformation T?
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What is an eigenspace?
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What is an eigenvalue?
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Define an eigenvector.
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What is the Gram-Schmidt process?
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What is a homogeneous equation?
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What is the image of a vector x under a transformation T?
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An inconsistent system has at least one solution.
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What is an indefinite quadratic form?
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What does it mean for a vector space to be infinite-dimensional?
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What is the inverse of a matrix?
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Define an invertible linear transformation.
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What is an invertible matrix?
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What are isomorphic vector spaces?
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Define isomorphism.
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What is the kernel of a linear transformation?
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What is the least-squares error?
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What is a least-squares solution?
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How is length or norm of a vector defined?
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What is a linear combination?
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A set of vectors is linearly dependent if it has only the trivial solution.
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A set of vectors is linearly independent if the only solution to their linear combination equaling zero is the trivial solution.
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What is a linear transformation T?
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What is a negative definite quadratic form?
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Define a quadratic form.
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What is the range of a linear transformation T?
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What is the rank of a matrix A?
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What does it mean for matrices to be row equivalent?
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What is the row space of a matrix A?
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What are similar matrices?
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What does Span{v1,...,vP} represent?
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Define a symmetric matrix.
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What is a subspace?
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What is the trace of a square matrix A?
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What is a unique solution?
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Study Notes
Key Concepts in Linear Algebra
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Algebraic Multiplicity: Refers to the number of times an eigenvalue appears as a root of the characteristic equation.
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Basic Variable: A variable corresponding to a pivot column in the coefficient matrix of a linear system, crucial for finding solutions.
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Basis: A maximal linearly independent set that spans a subspace H, represented as B = {v1,…,vP}.
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Best Approximation: The closest point in a subspace to a given vector, key in optimization problems.
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Characteristic Equation: Defined as det(A-λI) = 0, it is critical in finding eigenvalues of a matrix.
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Codomain: The set R^m that includes the range of a transformation T; if T maps V to W, then W is the codomain.
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Column Space: The set Col A comprises all linear combinations of the matrix A's columns, given as Col A = Span {a1,…,aN}.
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Consistent Linear System: A linear system that has at least one solution, vital for determining solvability.
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Diagonalizable Matrix: A matrix that can be expressed as PDP⁻¹, where D is a diagonal matrix and P is invertible.
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Diagonal Matrix: A square matrix with all non-diagonal entries equal to zero, simplifying calculations.
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Dimension of a Subspace: Indicates the number of vectors in a basis for subspace S, providing insight into its size.
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Domain of a Transformation: The set of all vectors x for which T(x) is defined, essential in understanding transformer mappings.
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Eigenspace: Comprises all solutions of Ax = λx for a given eigenvalue λ, including zero vector and all corresponding eigenvectors.
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Eigenvalue: A scalar λ such that the equation Ax = λx has a solution for some nonzero vector x, fundamental in spectral theory.
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Eigenvector: A nonzero vector x that satisfies Ax = λx for some scalar λ, representing directions scaled by transformations.
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Gram-Schmidt Process: An algorithm to create an orthogonal or orthonormal basis from a set of vectors, ensuring minimal redundancy.
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Homogeneous Equation: Represented as Ax = 0, this form is pivotal in analyzing linear systems.
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Image of a Vector: The vector T(x) resulting from applying transformation T to vector x, showcasing how transformations operate.
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Inconsistent System: A linear system with no solutions, indicating contradictions within the system.
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Indefinite Quadratic Form: A quadratic form that can take both positive and negative values, influencing optimization and geometry.
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Infinite-Dimensional Vector Space: A vector space V that lacks a finite basis, significant in advanced functional analysis.
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Inverse of a Matrix: A matrix A possesses an inverse A⁻¹, satisfying AA⁻¹ = A⁻¹A = I, critical in solving linear systems.
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Invertible Linear Transformation: A transformation T: R^N → R^N where an inverse function S exists, critical for establishing bijuctions.
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Invertible Matrix: A square matrix that has an inverse, a requisite for many operations in linear algebra.
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Isomorphic Vector Spaces: Two spaces V and W that can be connected through a one-to-one linear transformation T, preserving structure.
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Isomorphism: A one-to-one linear mapping between two vector spaces that allows for structural equivalence.
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Kernel of a Linear Transformation: The set of all vectors in V mapped to the zero vector, essential for understanding dimensions of transformations.
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Least-Squares Error: The distance ‖b - Ax∧‖ from the vector b to the least-squares solution Ax∧, important in regression analysis.
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Least-Squares Solution: A vector x∧ that minimizes the residual error from Ax = b, central in approximation methods.
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Length or Norm of a Vector: Defined as ‖v‖ = √(v^Tv), measuring the magnitude of a vector.
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Linear Combination: A combination of vectors obtained by scaling and adding them, foundational to understanding vector spaces.
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Linearly Dependent Set: A set of vectors having a nontrivial solution to the equation c1v1 + ... + cPvP = 0, indicating redundancy.
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Linearly Independent Set: A set of vectors where the only solution to c1v1 + ... + cPvP = 0 is the trivial solution, indicating uniqueness.
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Linear Transformation: A function T that maps between vector spaces preserving structure, obeying specific linearity properties.
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Negative Definite Quadratic Form: A quadratic form that is negative for all nonzero x, indicating a specific curvature in geometric applications.
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Quadratic Form: Defined by Q(x) = x^(T)Ax, involving symmetric matrices, important in optimization and statistics.
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Range of a Linear Transformation: Comprises all vectors produced by T(x) for vectors x in its domain, crucial in understanding outputs.
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Rank of a Matrix: Indicates the dimension of the column space, providing insight into the solutions of a system.
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Row Equivalent Matrices: Matrices connected by a series of row operations, showing how transformations can yield equivalent systems.
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Row Space: The set of all linear combinations of the rows of a matrix, similarly characterized as Col A^T.
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Similar Matrices: Matrices related by A = PBP⁻¹ where P is invertible, important in studying matrix properties.
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Span: The collection of all linear combinations of a set of vectors, representing the subspace generated by those vectors.
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Symmetric Matrix: A matrix A where A^T = A, characterized by reflective properties and simplifying many calculations.
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Subspace: A subset H of vector space V that contains the zero vector and is closed under addition and scalar multiplication.
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Trace of a Square Matrix: The sum of the diagonal entries, providing insight into the matrix's properties.
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Unique Solution: The only solution to a system, indicating non-redundancy and specificity in linear systems.
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