Glossary

13.1 Vector spaces and linear transformations

3 min read•august 7, 2024

Vector spaces and linear transformations are the building blocks of linear algebra. They provide a framework for understanding multidimensional systems and their relationships. This topic lays the groundwork for more advanced concepts in matrix operations and linear equations.

These concepts are crucial for solving real-world problems in physics, engineering, and data science. Understanding vector spaces and linear transformations helps us model complex systems and analyze their behavior efficiently.

Vector Spaces

Definition and Properties

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  • consists of a set of vectors and two operations (addition and scalar multiplication) that satisfy certain properties
    • under addition and scalar multiplication
    • of addition and scalar multiplication
    • of addition
    • Existence of (zero vector for addition and scalar identity for scalar multiplication)
    • Existence of (additive inverse for each vector)
    • of scalar multiplication over vector addition and field addition
  • expresses a vector as a sum of scalar multiples of other vectors
    • v=a1v1+a2v2+...+anvnv = a_1v_1 + a_2v_2 + ... + a_nv_n, where v,v1,v2,...,vnv, v_1, v_2, ..., v_n are vectors and a1,a2,...,ana_1, a_2, ..., a_n are scalars
  • means a set of vectors cannot be expressed as linear combinations of each other
    • If a1v1+a2v2+...+anvn=0a_1v_1 + a_2v_2 + ... + a_nv_n = 0 has only the trivial solution a1=a2=...=an=0a_1 = a_2 = ... = a_n = 0, then {v1,v2,...,vn}\{v_1, v_2, ..., v_n\} is linearly independent

Basis and Dimension

  • is a linearly independent set of vectors that spans the entire vector space
    • Every vector in the space can be uniquely expressed as a linear combination of basis vectors
    • Example: Standard basis for R2\mathbb{R}^2 is {(1,0),(0,1)}\{(1, 0), (0, 1)\}
  • is the number of vectors in a basis for the vector space
    • All bases for a vector space have the same number of vectors
    • Example: Rn\mathbb{R}^n has dimension nn because its standard basis consists of nn vectors

Linear Transformations

Definition and Properties

  • is a function T:V→WT: V \to W between vector spaces VV and WW that preserves vector addition and scalar multiplication
    • T(u+v)=T(u)+T(v)T(u + v) = T(u) + T(v) for all u,v∈Vu, v \in V
    • T(cv)=cT(v)T(cv) = cT(v) for all v∈Vv \in V and scalar cc
  • (or null space) of a linear transformation T:V→WT: V \to W is the set of all vectors in VV that map to the zero vector in WW
    • ker(T)={v∈V:T(v)=0}\text{ker}(T) = \{v \in V : T(v) = 0\}
    • Kernel is a subspace of the domain VV
  • (or range) of a linear transformation T:V→WT: V \to W is the set of all vectors in WW that are outputs of TT
    • im(T)={T(v):v∈V}\text{im}(T) = \{T(v) : v \in V\}
    • Image is a subspace of the codomain WW

Isomorphism

  • is a bijective (one-to-one and onto) linear transformation between vector spaces
    • Preserves vector space structure
    • Implies the vector spaces have the same dimension
  • Example: The T:Rn→RnT: \mathbb{R}^n \to \mathbb{R}^n defined by an invertible n×nn \times n matrix is an isomorphism
    • Bijective because the matrix is invertible
    • Linear because matrix multiplication satisfies linearity properties

Eigenvalues and Eigenvectors

Definition and Properties

  • of a linear transformation T:V→VT: V \to V (or a square matrix AA) is a scalar λ\lambda such that there exists a non-zero vector vv satisfying T(v)=λvT(v) = \lambda v (or Av=λvAv = \lambda v)
    • vv is called an corresponding to the eigenvalue λ\lambda
    • Eigenvalues are roots of the characteristic polynomial det⁡(A−λI)=0\det(A - \lambda I) = 0
  • Eigenvectors corresponding to distinct eigenvalues are linearly independent
    • Eigenvectors form a basis for the vector space if the matrix is diagonalizable
  • Example: For the matrix A=(2112)A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}, the eigenvalues are λ1=3\lambda_1 = 3 and λ2=1\lambda_2 = 1, with corresponding eigenvectors v1=(1,1)v_1 = (1, 1) and v2=(1,−1)v_2 = (1, -1)

Key Terms to Review (20)

Associativity: Associativity is a fundamental property of binary operations that states the way in which operations are grouped does not affect the outcome. This means that when performing an operation on three elements, the result will be the same regardless of how the elements are paired. Understanding associativity is crucial because it helps simplify expressions and ensures consistent results across different scenarios in various mathematical structures.
Basis: A basis is a set of vectors in a vector space that are linearly independent and span the entire space. This means that any vector in the space can be expressed as a unique linear combination of the basis vectors. Understanding a basis is essential because it helps in simplifying complex vector operations and provides a framework for analyzing linear transformations.
Closure: Closure refers to the property of a set that ensures the result of a specified operation on elements of the set always produces an element that is also within the same set. This concept is fundamental in understanding how mathematical structures operate, as it guarantees that the application of operations does not lead to elements outside the original set. Recognizing closure helps to classify sets and operations, linking them to group structures and vector spaces.
Commutativity: Commutativity is a fundamental property of certain mathematical operations where the order of the operands does not affect the result. This property is crucial in various mathematical structures, including vector spaces, as it allows for the rearrangement of elements in expressions without changing their outcome. Commutativity applies to both addition and multiplication operations in vector spaces, making it easier to manipulate vectors and perform linear transformations.
Determinant: A determinant is a scalar value that can be computed from the elements of a square matrix and encapsulates important properties of the matrix, such as whether it is invertible and how it transforms space. The value of the determinant can indicate whether a system of linear equations has a unique solution, be used to find eigenvalues in characteristic equations, and reflect the behavior of linear transformations in vector spaces.
Dimension: Dimension refers to the number of independent directions in which a vector space can be spanned. It is a critical concept that helps in understanding the structure of vector spaces and linear transformations, indicating how many vectors are needed to form a basis for the space. The dimension provides insight into the capacity of a vector space to accommodate vectors and ultimately plays a key role in defining the properties and behavior of linear transformations.
Distributivity: Distributivity is a fundamental property in mathematics that allows for the multiplication of a number by a sum to be distributed across the terms of the sum. In the context of vector spaces and linear transformations, this property is crucial because it ensures that when scaling vectors or applying linear transformations, the operation behaves predictably and consistently across combinations of vectors. This property supports the structure of vector spaces by ensuring that linear combinations and transformations adhere to a coherent mathematical framework.
Eigenvalue: An eigenvalue is a scalar value associated with a linear transformation represented by a matrix, which describes how a vector is stretched or compressed during that transformation. When a matrix acts on an eigenvector, the output is simply the eigenvector multiplied by the eigenvalue, revealing deep insights into the structure of the transformation and its effects on vector spaces. This concept is crucial in solving differential equations, particularly in contexts involving specific boundary conditions and stability analysis.
Eigenvector: An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied to it. In other words, if you multiply a matrix by an eigenvector, the result is simply the eigenvector scaled by a certain value known as the eigenvalue. This relationship is key to understanding how linear transformations act within vector spaces and helps reveal the structure of those spaces.
Identity elements: Identity elements are special elements in algebraic structures, such as groups or vector spaces, that leave other elements unchanged when combined with them. They play a crucial role in defining operations and ensuring that certain properties hold, like closure and the existence of inverses. In the context of vector spaces and linear transformations, the identity element ensures that the action of a transformation doesn't alter the vector it is applied to.
Image: In the context of vector spaces and linear transformations, the image refers to the set of all output vectors that can be produced by applying a linear transformation to input vectors from the domain. This concept highlights how a transformation maps elements from one space to another, emphasizing the relationship between input and output through the rules of linearity. Understanding the image is crucial for analyzing properties such as dimensionality, rank, and the overall behavior of linear transformations.
Inverse elements: Inverse elements are elements in a mathematical structure that, when combined with a given element using a specific operation, yield an identity element for that operation. In the context of vector spaces and linear transformations, inverse elements are crucial for understanding how transformations can be reversed, leading to concepts such as linear independence and spanning sets.
Invertible matrix: An invertible matrix, also known as a non-singular matrix, is a square matrix that possesses an inverse. This means that there exists another matrix such that when it is multiplied by the original matrix, the result is the identity matrix. The existence of an inverse is closely linked to properties like determinants, rank, and linear independence, making it a crucial concept in understanding linear transformations and vector spaces.
Isomorphism: Isomorphism is a mathematical concept that describes a structural similarity between two objects, meaning there exists a one-to-one correspondence that preserves operations between them. This concept is crucial in understanding how different algebraic structures, like groups or vector spaces, can be equivalent in a certain sense. It helps in identifying when two systems can be treated as the same for practical purposes, even if their elements and operations appear different at first glance.
Kernel: The kernel of a linear transformation is the set of all input vectors that map to the zero vector in the output space. This concept is crucial because it helps us understand the properties of linear transformations and their behavior with respect to vector spaces. The kernel is a fundamental aspect of both the transformation and the structure of the vector space, revealing insights about solutions to linear equations and dimensionality.
Linear Combination: A linear combination is an expression formed by multiplying each vector in a set by a scalar and then adding the results together. This concept is foundational in understanding how vectors can be combined to create new vectors and is essential for exploring the structure of vector spaces and the solutions of systems of linear equations.
Linear independence: Linear independence refers to a set of vectors in a vector space where no vector can be expressed as a linear combination of the others. This concept is crucial as it helps determine the dimensionality of a vector space and informs us about the uniqueness of solutions in systems of linear equations.
Linear transformation: A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. It takes a vector as input and outputs another vector, maintaining the structure of the space, which makes it essential in understanding how different mathematical objects interact. These transformations can be represented using matrices, allowing for simpler calculations and deeper insights into their properties across various mathematical contexts.
Matrix transformation: A matrix transformation is a function that takes a vector as input and produces another vector by multiplying it with a matrix. This process is essential for understanding how linear transformations can be represented in a more manageable form, allowing for the manipulation and analysis of geometric and algebraic properties in vector spaces.
Vector Space: A vector space is a mathematical structure formed by a collection of vectors, which can be added together and multiplied by scalars. This concept is foundational in linear algebra, as it allows for the examination of linear combinations and transformations, making it essential for understanding various mathematical frameworks and physical theories. Within this structure, operations such as addition and scalar multiplication follow specific rules, providing a systematic way to approach problems in many areas of mathematics and physics.
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