Questions and Answers
What is the algebraic multiplicity?
The multiplicity of an eigenvalue as a root of the characteristic equation.
Which equation represents the associative law of multiplication?
What is an augmented matrix?
A matrix made up of a coefficient matrix for a linear system and one or more columns to the right.
What is an auxiliary equation?
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Define a basic variable.
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What is a basis in vector space?
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What is a change of coordinate matrix?
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Define a minimal spanning set.
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What is a non-homogeneous equation?
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What does it mean if a matrix is nonsingular?
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What is a nontrivial solution?
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Define the null space of a matrix.
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What is the parametric equation of a line?
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What is a pivot in matrix operations?
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What does the product Ax represent?
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Define the range of a linear transformation.
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What is the rank of a matrix?
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What is the row space of a matrix?
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Define the set spanned by {v1....vp}.
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What is a singular matrix?
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What is a solution set in the context of linear systems?
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What does span{v1....vp} refer to?
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Define a spanning set.
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What is meant by standard basis?
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What is a standard matrix?
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What is a subspace?
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What is a system of linear equations?
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What is a transformation function?
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What is a transpose of a matrix?
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Define a vector.
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What is a vector equation?
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What is a vector space?
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What is a zero subspace?
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What is a zero vector?
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Study Notes
Algebraic Concepts
- Algebraic Multiplicity: Refers to the number of times an eigenvalue appears as a root in the characteristic equation of a matrix.
- Associative Law of Multiplication: Indicates the grouping of multiplication does not affect the final product, expressed as A(BC) = (AB)C.
- Augmented Matrix: Combines the coefficient matrix of a linear system with additional columns to the right, representing constants or additional variables.
Equations and Systems
- Auxiliary Equation: A polynomial generated from the coefficients of a homogeneous difference equation, used in solving these equations.
- Non-Homogeneous Equation: An equation in the form Ax = b where b is not equal to zero, indicating the presence of a constant term.
Linear Systems Variables
- Basic Variable: A variable corresponding to a pivot column in the coefficient matrix, crucial for solving linear systems.
- Non-Trivial Solution: A solution to a homogeneous equation that is not zero, highlighting the existence of alternative solutions.
Vector Spaces and Sets
- Basis: An indexed linearly independent set that spans a vector space, forming the foundation for the space.
- Null Space: The collection of solutions to the homogeneous equation Ax = 0, defining the kernel of the matrix A.
- Range: The set of all vectors produced by the transformation T(x) for inputs x in its domain.
- Row Space: Consists of linear combinations of the rows of a matrix, also known as Col A^T.
Matrix Properties
- Nonsingular Matrix: Indicates a matrix that is invertible, meaning it has a unique inverse.
- Singular Matrix: A square matrix lacking an inverse, rendering it non-invertible.
- Standard Matrix: Represents the linear transformation T(x) = Ax for all x in the domain, allowing for transformation representation.
Span and Linear Combinations
- Spanning Set: Any collection of vectors that can generate the entirety of a vector space through linear combinations.
- Set Spanned by {v1....vp}: The span represents all possible linear combinations of the vectors in the set.
- Minimal Spanning Set: A spanning set where removal of any single element prevents it from spanning the original space.
Transformations and Operations
- Transformation Function: A function that assigns each vector x in Rn a corresponding unique vector T(x) in Rm.
- Vector Equation: An equation expressing a linear combination of vectors with undetermined coefficients, vital in solving systems or transformations.
Additional Concepts
- Zero Vector: The unique vector where all components are zero, serving as the identity element in vector addition.
- Zero Subspace: A special subspace consisting solely of the zero vector, representing the least dimension in vector spaces.
- Parametric Equation of Line: Expressed as x = p + tv, describing a line in terms of a point p and a direction vector v.
- Rank: Defined as the dimension of the column space of a matrix, playing a crucial role in determining the solutions of linear systems.
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Test your knowledge of key terms in linear algebra with these flashcards. Each card features an important word along with its definition, helping you to better understand the concepts used in the subject. Perfect for students looking to reinforce their vocabulary related to linear algebra.