Linear Algebra 1: Definitions and Theorems Flashcards

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Questions and Answers

What is a linear combination?

A set of vectors

A sum of scalar multiples of vectors (correct)

A linear equation

An array of numbers

What characterizes a linear equation?

A linear combination that sums to some constant, with a solution set of vectors.

Define homogeneous linear equation.

A linear equation with a constant of zero.

What is Gauss' method?

A method involving switching equations, multiplying by a non-zero scalar, or replacing an equation by the sum of itself and another scaled equation. Signup and view all the answers

What defines a vector space?

A set of vectors that is closed under addition and scalar multiplication, contains the zero vector, and satisfies other properties. Signup and view all the answers

What is a trivial space?

A one-element vector space. Signup and view all the answers

State the 0 vector space lemma.

0v = 0, (-1v) + v = 0, r * 0 = 0. Signup and view all the answers

What is a subspace?

A subset of a vector space that itself is a vector space. Signup and view all the answers

What does span mean in linear algebra?

The set of all linear combinations of vectors for all non-empty subsets of a vector space. Signup and view all the answers

What is the span lemma?

In a vector space, the span of any subset is a subspace. Signup and view all the answers

Define linear independence.

A set of vectors where none are a linear combination of the others in that set. Signup and view all the answers

What is linear dependence?

A set of vectors where at least one vector is a linear combination of the others. Signup and view all the answers

What does the linear independence lemma state?

A subset is linearly independent if the only linear relationship among its elements equates to the zero vector when all scalars are zero. Signup and view all the answers

What is a leading variable?

The first variable in a row with a nonzero constant. Signup and view all the answers

What is echelon form?

A linear system where the leading variable is to the right of the leading variable in the row above. Signup and view all the answers

What are row operations?

Operations include swapping rows, multiplying a row by a scalar, and row combination. Signup and view all the answers

What is the solution set of a linear system?

A linear combination of the particular solution and all free variables with their associated vectors. Signup and view all the answers

Define free variable.

Any non-leading, non-zero variable when in echelon form. Signup and view all the answers

What does singular refer to in linear algebra?

A matrix of coefficients of a homogeneous system with infinitely many solutions. Signup and view all the answers

What does nonsingular refer to?

Matrix coefficients of a homogeneous system with a unique solution. Signup and view all the answers

What characterizes a row reduced echelon form?

A matrix in echelon form where all leading variables are 1 and are the only non-zero entries in their columns. Signup and view all the answers

What does the reversibility lemma state?

All elementary row operations are reversible. Signup and view all the answers

What does the 'reduces to' lemma describe?

'Reduces to' between matrices is an equivalence relation. Signup and view all the answers

Define row equivalence.

When two matrices are inter-reducible via elementary row operations. Signup and view all the answers

What does the linear combination lemma state?

A linear combination of a linear combination is a linear combination. Signup and view all the answers

What does the row reduced echelon form theorem state?

Each matrix is row equivalent to a unique row reduced echelon form matrix. Signup and view all the answers

Define a vector.

An Mx1 matrix with entries as components. Signup and view all the answers

What is the solution to a linear equation?

A set of vectors that provides a solution to a linear system. Signup and view all the answers

What is a vector sum?

The addition of two vectors where the components of each are added according to row. Signup and view all the answers

What is scalar multiplication?

The multiplication of a vector by a scalar where each component is multiplied by the scalar. Signup and view all the answers

Define a matrix.

A M rows by N columns array of numbers. Signup and view all the answers

What is an entry in a matrix?

A number in a matrix. Signup and view all the answers

What does the set dependence lemma state?

A subset is linearly independent if its set is linear. Signup and view all the answers

Study Notes

Linear Algebra Concepts

Linear Combination: A representation of a vector as a sum of scalar multiples of other vectors. Formulated as (a_1x_1 + a_2x_2 + ... + a_nx_n).
Linear Equation: A specific type of linear combination that equals a constant (d), expressed as (a_1x_1 + a_2x_2 + ... + a_nx_n = d).
Homogeneous Equation: A linear equation where the constant is zero, characterized by (a_1x_1 + a_2x_2 + ... + a_nx_n = 0).
Gauss' Method: Techniques for manipulating equations, including: switching two equations, multiplying an equation by a non-zero scalar, and replacing an equation with a scaled sum of itself and another equation.
Vector Space: A collection of vectors satisfying these conditions: closed under addition and scalar multiplication, contains the zero vector, commutativity, associativity, existence of additive inverses, and identity for scalar multiplication.
Trivial Space: A vector space consisting solely of the zero vector, containing only one element.
0 Vector Space Lemma: Establishes properties such as (0 \cdot v = 0), ((-1 \cdot v) + v = 0), and (r \cdot 0 = 0).
Subspace: A subset of a vector space that meets all the vector space criteria itself.
Span: The complete set of possible linear combinations of a subset of vectors from a vector space.
Span Lemma: This asserts that the span of any subset of a vector space constitutes a subspace.
Linear Independence: Denotes a set of vectors where no vector is a linear combination of others in the set.
Linear Dependence: Describes a situation where at least one vector in the set can be represented as a linear combination of others.
Linear Independence Lemma: A characterization of linear independence; a subset is independent if the only solution to the equation (c_1s_1 + c_2s_2 + ... + c_ns_n = 0) is when all coefficients (c_i) are zero.

Matrix Operations

Leading Variable: The first non-zero variable in any row of a matrix.
Echelon Form: A matrix form where leading variables are consistently to the right of leading variables in preceding rows, maintaining a triangular structure.
Row Operations: Fundamental operations for manipulating matrices: swapping rows, multiplying a row by a non-zero scalar, and adding a scaled row to another.
Linear System Solution Set: The complete solution representation, incorporating particular solutions and free variables.
Free Variable: A variable in echelon form that does not serve as a leading variable.
Singular Matrix: A coefficient matrix of a homogeneous system that possesses infinitely many solutions.
Nonsingular Matrix: A coefficient matrix that yields a unique solution for a homogeneous system.
Row Reduced Echelon Form: A modified matrix form where each leading variable is isolated and equals 1, while all other entries in the column are zero.
Reversibility Lemma: Indicates that all elementary row operations can be reversed.
'Reduces to' Lemma: Establishes an equivalence relation among matrices concerning reducibility via elementary operations.
Row Equivalence: A condition where two matrices can be converted into one another through a series of elementary row operations.

Additional Concepts

Linear Combination Lemma: States that a linear combination of linear combinations is still a linear combination.
Row Reduced Echelon Form Theorem: Asserts that every matrix is row equivalent to a unique row reduced echelon form.
Vector: An Mx1 matrix denoted by its components.
Solution to a Linear Equation: Represents the set of vectors that satisfy a linear system.
Vector Sum: The component-wise addition of two vectors.
Scalar Multiplication: The multiplication of every component of a vector by a scalar value.
Matrix: An array of numbers arranged in M rows and N columns.
Entry: Any individual number or element contained within a matrix.
Set Dependence Lemma: Refers to the concept of linear independence in a subset context.

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Test your understanding of key concepts in Linear Algebra with these flashcards. Covering definitions, lemmas, and theorems, this quiz is essential for mastering the fundamentals of linear combinations, linear equations, and homogeneous systems. Perfect for students looking to solidify their knowledge in this crucial area of mathematics.