Glossary

1.3 Linear Combinations and Linear Independence

4 min read•july 30, 2024

Linear combinations and independence are key concepts in vector spaces. They help us understand how vectors relate to each other and form subspaces. These ideas are crucial for grasping the structure of vector spaces and their subspaces.

By exploring linear combinations, we can see how vectors can be built from others. Linear independence shows which vectors are truly unique. These concepts lay the groundwork for understanding bases and dimensions in vector spaces.

Linear Combinations of Vectors

Definition and Properties

Top images from around the web for Definition and Properties

  • OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Finding Linear Equations View original

    Is this image relevant?

  • A theorem in Linear Algebra; linear dependence - Axler - Mathematics Stack Exchange View original

    Is this image relevant?

  • OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Finding Linear Equations View original

    Is this image relevant?

1 of 3

Top images from around the web for Definition and Properties

  • OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Finding Linear Equations View original

    Is this image relevant?

  • A theorem in Linear Algebra; linear dependence - Axler - Mathematics Stack Exchange View original

    Is this image relevant?

  • OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials: Finding Linear Equations View original

    Is this image relevant?

1 of 3
  • A of vectors is a sum of scalar multiples of the vectors
    • For vectors v1,v2,...,vnv_1, v_2, ..., v_n in a vector space VV and scalars c1,c2,...,cnc_1, c_2, ..., c_n in a field FF, a linear combination is c1v1+c2v2+...+cnvnc_1v_1 + c_2v_2 + ... + c_nv_n
    • The scalars in a linear combination are called the coefficients
  • The set of all possible linear combinations of a given set of vectors forms a subspace of the vector space
  • If the zero vector can be expressed as a linear combination of a set of vectors with not all coefficients being zero, then the set is

Examples and Applications

  • A linear combination of two vectors v1=(1,2)v_1 = (1, 2) and v2=(3,4)v_2 = (3, 4) with coefficients c1=2c_1 = 2 and c2=−1c_2 = -1 is 2v1−v2=(2,4)−(3,4)=(−1,0)2v_1 - v_2 = (2, 4) - (3, 4) = (-1, 0)
  • In physics, the resultant force acting on an object can be expressed as a linear combination of the individual forces acting on it
  • In computer graphics, points on a 3D surface can be represented as linear combinations of the vertices defining the surface

Expressing Vectors as Linear Combinations

Solving for Coefficients

  • To determine if a vector vv can be expressed as a linear combination of vectors v1,v2,...,vnv_1, v_2, ..., v_n, solve the equation c1v1+c2v2+...+cnvn=vc_1v_1 + c_2v_2 + ... + c_nv_n = v for the coefficients c1,c2,...,cnc_1, c_2, ..., c_n
    • If a solution exists for the coefficients, then vv is a linear combination of the given vectors
    • If the only solution for the coefficients is the trivial solution (all coefficients are zero), then vv is not a linear combination of the given vectors
  • The process of solving for the coefficients involves setting up a system of linear equations and using techniques such as Gaussian elimination or matrix inversion

Examples and Applications

  • To express the vector (2,3)(2, 3) as a linear combination of the vectors (1,1)(1, 1) and (1,2)(1, 2), solve the equation c1(1,1)+c2(1,2)=(2,3)c_1(1, 1) + c_2(1, 2) = (2, 3) for c1c_1 and c2c_2. The solution is c1=1c_1 = 1 and c2=1c_2 = 1, so (2,3)=1(1,1)+1(1,2)(2, 3) = 1(1, 1) + 1(1, 2)
  • In cryptography, a message can be encoded as a linear combination of vectors, and decoding involves expressing the encoded message as a linear combination of the same basis vectors

Linear Independence and Dependence

Definitions and Properties

  • A set of vectors is if no vector in the set can be expressed as a linear combination of the other vectors in the set
  • A set of vectors is linearly dependent if at least one vector in the set can be expressed as a linear combination of the other vectors in the set
    • For a set of vectors v1,v2,...,vnv_1, v_2, ..., v_n, the set is linearly dependent if there exist scalars c1,c2,...,cnc_1, c_2, ..., c_n, not all zero, such that c1v1+c2v2+...+cnvn=0c_1v_1 + c_2v_2 + ... + c_nv_n = 0
    • If the only solution to the equation c1v1+c2v2+...+cnvn=0c_1v_1 + c_2v_2 + ... + c_nv_n = 0 is the trivial solution (all coefficients are zero), then the set of vectors is linearly independent
  • The zero vector is always linearly dependent on any set of vectors
  • A set containing the zero vector is always linearly dependent

Examples and Applications

  • The set of vectors {(1,0),(0,1)}\{(1, 0), (0, 1)\} is linearly independent in R2\mathbb{R}^2, as neither vector can be expressed as a scalar multiple of the other
  • The set of vectors {(1,2),(2,4),(3,6)}\{(1, 2), (2, 4), (3, 6)\} is linearly dependent, as (3,6)=1(1,2)+1(2,4)(3, 6) = 1(1, 2) + 1(2, 4)
  • In quantum mechanics, a set of quantum states is linearly independent if no state can be expressed as a linear combination of the others

Identifying Linear Independence vs Dependence

Solving the Homogeneous Equation

  • To determine if a set of vectors v1,v2,...,vnv_1, v_2, ..., v_n is linearly independent or dependent, solve the equation c1v1+c2v2+...+cnvn=0c_1v_1 + c_2v_2 + ... + c_nv_n = 0 for the coefficients c1,c2,...,cnc_1, c_2, ..., c_n
    • If the only solution is the trivial solution (all coefficients are zero), then the set is linearly independent
    • If there exists a non-trivial solution (at least one coefficient is non-zero), then the set is linearly dependent
  • The process of solving for the coefficients involves setting up a system of linear equations and using techniques such as Gaussian elimination or matrix inversion

Additional Considerations

  • If the number of vectors in the set is greater than the of the vector space, then the set is necessarily linearly dependent
  • If a subset of a linearly independent set is removed, the remaining set is still linearly independent
  • In Rn\mathbb{R}^n, a set of nn vectors is linearly independent if and only if the determinant of the matrix formed by the vectors as columns is non-zero

Key Terms to Review (18)

Basis: A basis is a set of vectors in a vector space that is linearly independent and spans the entire space. It provides a way to express any vector in the space as a linear combination of the basis vectors, establishing a framework for understanding the structure and dimensions of vector spaces.
Basis Theorem: The Basis Theorem states that any vector space has a basis, which is a set of linearly independent vectors that spans the space. This means that every vector in the space can be expressed as a linear combination of the basis vectors. The significance of this theorem lies in its ability to define the dimensionality of the space and understand the structure of vector spaces.
Column Space: The column space of a matrix is the set of all possible linear combinations of its column vectors, essentially representing all vectors that can be formed using those columns. This concept is crucial for understanding how matrices transform input vectors and plays a key role in identifying solutions to systems of linear equations. The column space is a type of subspace within the larger vector space, making it vital for grasping ideas around span, independence, and dimensions in linear algebra.
Dimension: Dimension refers to the number of vectors in a basis for a vector space, which essentially measures the 'size' or 'degrees of freedom' of that space. Understanding dimension is crucial for grasping concepts like subspaces, linear combinations, and bases, as it helps in recognizing how many independent directions or parameters can exist in that space.
Euclidean Space: Euclidean space is a mathematical construct that provides a framework for understanding geometric relationships in two or more dimensions, characterized by the familiar concepts of points, lines, and planes. It serves as the foundation for vector spaces, allowing us to perform operations such as addition and scalar multiplication while maintaining the essential geometric properties. This space is integral to understanding linear combinations, independence, finite dimensions, and concepts of orthogonality, forming a cornerstone in many areas of mathematics.
Function Space: A function space is a collection of functions that share a common property, often structured as a vector space itself. This means that within this space, you can perform operations like addition and scalar multiplication on functions, making it possible to analyze them using the principles of linear algebra. Understanding function spaces helps in grasping how functions can be combined and their linear independence, which is crucial in various mathematical contexts.
Generating Set: A generating set is a collection of vectors in a vector space such that every vector in that space can be expressed as a linear combination of the vectors in the set. This concept is crucial as it helps to understand how entire vector spaces can be constructed using smaller sets of vectors. The generating set provides insights into the structure of the vector space, including its dimensions and properties like linear independence and span.
Linear Combination: A linear combination is an expression formed by multiplying each vector in a set by a corresponding scalar and then adding the results. This concept is foundational in understanding how vectors can be combined to create new vectors, which is crucial for exploring subspaces, spans, and linear independence within vector spaces.
Linearly dependent: Linearly dependent refers to a set of vectors in which at least one vector can be expressed as a linear combination of the others. This means that there exists a non-trivial solution to the equation formed by setting the linear combination of these vectors equal to zero. In this context, understanding linear dependence is crucial for analyzing the relationships between vectors and determining their independence or redundancy in spanning a vector space.
Linearly independent: Linearly independent refers to a set of vectors in a vector space that cannot be expressed as a linear combination of each other. In simpler terms, this means no vector in the set can be created by combining the others with any coefficients. This concept is crucial for understanding the structure of vector spaces and determining the dimensions they occupy, as well as how many vectors are needed to span that space without redundancy.
Matrix Rank: Matrix rank refers to the maximum number of linearly independent column vectors (or row vectors) in a matrix, which provides crucial insight into the solutions of a system of linear equations. It indicates the dimension of the vector space generated by its columns or rows and reflects the matrix's ability to represent linear transformations effectively. Understanding the rank is essential for determining whether a system of equations has a unique solution, infinitely many solutions, or no solution at all.
Maximal linearly independent set: A maximal linearly independent set is a collection of vectors that is both linearly independent and cannot be extended by adding another vector without losing its independence. This means that no vector in the set can be expressed as a linear combination of the others, and adding any other vector from the space would create a dependency. Maximal sets are significant because they provide a basis for the vector space when considered with all linearly independent vectors.
No nontrivial solutions: The phrase 'no nontrivial solutions' refers to a situation in linear algebra where the only solution to a homogeneous linear equation or system is the trivial solution, which is typically when all variables are equal to zero. This concept is essential in understanding the structure of linear combinations and the implications for linear independence, as it indicates that the set of vectors involved does not allow for any other combinations to yield the zero vector without defaulting to zero coefficients.
Rank-Nullity Theorem: The rank-nullity theorem states that for a linear transformation from a finite-dimensional vector space to another, the sum of the rank and the nullity of the transformation equals the dimension of the domain. This theorem connects the concepts of linear combinations, independence, and the properties of transformations, establishing a fundamental relationship between the solutions to linear equations and their geometric interpretations.
Scalar multiplication: Scalar multiplication is an operation that takes a scalar (a single number) and a vector (or matrix) and produces another vector (or matrix) by multiplying each component by the scalar. This operation is fundamental in various mathematical contexts as it helps to stretch, shrink, or reverse the direction of vectors, thereby playing a critical role in the structure of vector spaces, linear combinations, and matrix operations.
Span: Span is the set of all possible linear combinations of a given set of vectors in a vector space. It helps define the extent to which a set of vectors can cover or represent other vectors within that space, playing a crucial role in understanding subspaces and dimensionality.
Unique Representation: Unique representation refers to the idea that a vector in a vector space can be expressed in exactly one way as a linear combination of a given set of vectors. This concept is closely tied to linear independence, as a set of vectors that allows for unique representation does not contain any redundant vectors, meaning no vector in the set can be written as a combination of the others. Understanding unique representation is crucial for grasping how vectors relate to each other and how they can form a basis for a vector space.
Vector Addition: Vector addition is the process of combining two or more vectors to form a new vector, representing the cumulative effect of the individual vectors. This operation is fundamental in vector spaces, as it adheres to specific axioms that govern the behavior of vectors, such as closure, associativity, and commutativity. The result of vector addition retains the properties necessary for defining linear combinations and understanding linear independence.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide