Linear Algebra Vocabulary Flashcards
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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?

  • A(BC)
  • A(BC) = (AB)C (correct)
  • (AB)C
  • A + B = C

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?

<p>A polynomial equation in a variable r, created from the coefficients of a homogeneous difference equation.</p> Signup and view all the answers

Define a basic variable.

<p>A variable in a linear system that corresponds to a pivot column in the coefficient matrix.</p> Signup and view all the answers

What is a basis in vector space?

<p>An indexed set B={v1.....vp} in V such that: B is a linearly independent set and H = span{v1...vp}.</p> Signup and view all the answers

What is a change of coordinate matrix?

<p>A matrix PC Ax where A is an m x n matrix and x represents any vector in Rn.</p> Signup and view all the answers

Define a minimal spanning set.

<p>A set B that spans H and has the property that if one of the elements is removed from B then the new set does not span H.</p> Signup and view all the answers

What is a non-homogeneous equation?

<p>Ax = b with b not equal to 0.</p> Signup and view all the answers

What does it mean if a matrix is nonsingular?

<p>It means the matrix is invertible.</p> Signup and view all the answers

What is a nontrivial solution?

<p>A nonzero solution of a homogeneous equation or system of homogeneous equations.</p> Signup and view all the answers

Define the null space of a matrix.

<p>The set of all solutions of the homogeneous equation Ax=0.</p> Signup and view all the answers

What is the parametric equation of a line?

<p>x = p + tv.</p> Signup and view all the answers

What is a pivot in matrix operations?

<p>A nonzero number that is used in a pivot position to create zeros through row operations.</p> Signup and view all the answers

What does the product Ax represent?

<p>The linear combination of the columns of A using the corresponding entries in x as weights.</p> Signup and view all the answers

Define the range of a linear transformation.

<p>The set of all vectors of the form T(x) for some x in the domain of T.</p> Signup and view all the answers

What is the rank of a matrix?

<p>The dimension of the column space of A, denoted as rank A.</p> Signup and view all the answers

What is the row space of a matrix?

<p>The set Row A of all linear combinations of the vectors formed from the rows of A.</p> Signup and view all the answers

Define the set spanned by {v1....vp}.

<p>The set Span{v1.....vp}.</p> Signup and view all the answers

What is a singular matrix?

<p>A square matrix that has no inverse.</p> Signup and view all the answers

What is a solution set in the context of linear systems?

<p>The set of all possible solutions of a linear system; the set is empty when the linear system is inconsistent.</p> Signup and view all the answers

What does span{v1....vp} refer to?

<p>The set of all linear combinations of v1.....vp.</p> Signup and view all the answers

Define a spanning set.

<p>Any set {v1....vp} in H such that H = span {v1.....vp}.</p> Signup and view all the answers

What is meant by standard basis?

<p>The basis E = {e1....en}.</p> Signup and view all the answers

What is a standard matrix?

<p>The matrix A such that T(x) = Ax for all x in the domain T.</p> Signup and view all the answers

What is a subspace?

<p>A subset H of some vector space V that is closed under vector addition and scalar multiplication and contains the zero vector.</p> Signup and view all the answers

What is a system of linear equations?

<p>A set of two or more equations with two or more variables.</p> Signup and view all the answers

What is a transformation function?

<p>A rule that assigns to each vector x in Rn a unique vector T(x) in Rm.</p> Signup and view all the answers

What is a transpose of a matrix?

<p>An n x m matrix A^T whose columns are the corresponding rows of the m x n matrix A.</p> Signup and view all the answers

Define a vector.

<p>A list of numbers or a matrix with only one column.</p> Signup and view all the answers

What is a vector equation?

<p>An equation involving a linear combination of vectors with undetermined weights.</p> Signup and view all the answers

What is a vector space?

<p>A set of objects, called vectors, on which two operations are defined, adhering to ten axioms.</p> Signup and view all the answers

What is a zero subspace?

<p>The subspace {0} consisting of only the zero vector.</p> Signup and view all the answers

What is a zero vector?

<p>The unique vector such that the entries are all zero.</p> Signup and view all the answers

Flashcards

Algebraic Multiplicity

The number of times an eigenvalue appears as a root in the characteristic equation of a matrix.

Associative Law of Multiplication

The order of multiplication does not affect the outcome: A(BC) = (AB)C

Augmented Matrix

A matrix combining the coefficient matrix of a linear system with constants or additional variables in columns to the right.

Auxiliary Equation

A polynomial equation defined by the coefficients of a homogeneous difference equation, used in solving these equations.

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Non-Homogeneous Equation

An equation with a constant term, not always equal to zero, represented by Ax = b where b ≠ 0.

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Basic Variable

A variable in a linear system corresponding to a pivot column in the coefficient matrix, crucial for solving the system.

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Non-Trivial Solution

A solution to a homogeneous equation, but not the zero vector, suggesting multiple possible solutions.

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Basis

A set of linearly independent vectors able to span the entirety of a vector space.

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Null Space

The set of all solutions to the homogeneous equation Ax = 0, representing the 'kernel' of the matrix A.

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Range

The set of all outputs from a linear transformation T(x) for inputs x in its domain.

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Row Space

Consists of linear combinations of the rows of a matrix, considered the column space of the matrix A^T.

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Nonsingular Matrix

A matrix that is invertible, meaning it has a unique inverse.

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Singular Matrix

A square matrix without an inverse, rendering it non-invertible.

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Standard Matrix

A matrix representation of a linear transformation T(x) = Ax for all x in the domain.

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Spanning Set

A collection of vectors that can create all vectors in a vector space through linear combinations.

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Set Spanned by {v1....vp}

The set of all possible linear combinations of the vectors in the set {v1....vp}.

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Minimal Spanning Set

A spanning set where removing any single element results in a set that can no longer span the original space.

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Transformation Function

A function that maps each vector x in Rn to a unique vector T(x) in Rm.

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Vector Equation

An equation representing a linear combination of vectors with undetermined coefficients, used in solving systems or for transformations.

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Zero Vector

The unique vector where all components are zero, serving as the identity element in vector addition.

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Zero Subspace

The subspace containing only the zero vector, representing the smallest possible dimension in vector spaces.

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Parametric Equation of Line

x = p + tv, where p is a point on the line and v is a direction vector.

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Rank

The dimension of the column space of a matrix, playing a role in determining solutions of linear systems.

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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.

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