Glossary

2.2 Kernel and range of a linear transformation

4 min read•august 16, 2024

Linear transformations are like magical machines that turn vectors into other vectors. The and are two key parts of these machines. They help us understand what goes in and what comes out.

The kernel tells us which vectors become zero, while the range shows all possible outputs. These concepts are super important for figuring out if a transformation is one-to-one or covers everything. They're the building blocks for understanding linear transformations.

Kernel and Range of Linear Transformations

Defining Kernel and Range

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  • Kernel () of encompasses vectors in V mapped to zero vector in W
  • Mathematical definition of kernel denoted as ker(T) = {v ∈ V | T(v) = 0}
  • Range () of linear transformation T: V → W includes all vectors in W that are outputs of T for some input vector in V
  • Mathematical definition of range denoted as range(T) = {w ∈ W | ∃v ∈ V, T(v) = w}
  • Both kernel and range form subspaces of their respective vector spaces
  • Dimension of kernel called of T, dimension of range called of T
  • states = + for linear transformation T: V → W

Properties and Relationships

  • Kernel provides information about preimage of zero vector in codomain
  • Range reveals set of all possible outputs of transformation
  • Nullity of equals zero
  • Rank of equals dimension of codomain
  • Relationship between kernel, range, and injectivity/surjectivity crucial for understanding isomorphisms between vector spaces
  • For finite-dimensional vector spaces, injectivity linked to full column rank of matrix representation
  • For finite-dimensional vector spaces, surjectivity linked to full row rank of matrix representation

Determining Kernel and Range

Finding the Kernel

  • Solve T(v) = 0 to determine kernel
  • For matrix transformations, solving kernel involves homogeneous system Ax = 0 (A represents T)
  • Use Gaussian elimination to find basis for kernel
  • Express solution as set of vectors
  • Verify nullity (dimension of kernel) adheres to Rank-Nullity Theorem

Determining the Range

  • Apply transformation to basis of domain
  • Express resulting vectors as spanning set
  • For matrix transformations, range equals column space of matrix A representing T
  • Use Gaussian elimination to determine dimension of range (rank)
  • Verify rank adheres to Rank-Nullity Theorem
  • Analyze relationship between domain, codomain, and range to understand transformation properties

Examples and Applications

  • Linear transformation T: R3→R2\mathbb{R}^3 \to \mathbb{R}^2 defined by T(x, y, z) = (x + y, y - z)
    • Kernel: solve T(x, y, z) = (0, 0) to get ker(T) = {(t, -t, -t) | t ∈ R\mathbb{R}}
    • Range: apply T to basis vectors (1, 0, 0), (0, 1, 0), (0, 0, 1) to get range(T) = span{(1, 1), (1, -1)}
  • Rotation matrix R = (cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}
    • Kernel: ker(R) = {(0, 0)} (only zero vector)
    • Range: range(R) = R2\mathbb{R}^2 (entire codomain)

Kernel and Injectivity

Relationship Between Kernel and Injectivity

  • Linear transformation T injective (one-to-one) if and only if kernel contains only zero vector
  • Non-injective T implies existence of distinct vectors v1 and v2 in V where T(v1) = T(v2)
  • Non-zero vector v1 - v2 in kernel indicates lack of injectivity
  • Injectivity determined by analyzing dimension of kernel or rank of transformation matrix
  • Nullity of injective linear transformation equals zero

Examples and Applications

  • Identity transformation I: V → V always injective (kernel contains only zero vector)
  • Projection transformation P: R3→R2\mathbb{R}^3 \to \mathbb{R}^2 defined by P(x, y, z) = (x, y) not injective
    • ker(P) = {(0, 0, t) | t ∈ R\mathbb{R}} (non-zero vectors in kernel)
  • Linear transformation T: R2→R3\mathbb{R}^2 \to \mathbb{R}^3 defined by T(x, y) = (x, y, x+y) injective
    • ker(T) = {(0, 0)} (only zero vector in kernel)

Range and Surjectivity

Relationship Between Range and Surjectivity

  • Linear transformation T surjective (onto) if and only if range equals codomain
  • Non-surjective T implies existence of vector w in W not in range of T
  • Surjectivity determined by comparing dimension of range to dimension of codomain
  • Rank of surjective linear transformation equals dimension of codomain

Examples and Applications

  • Linear transformation T: R3→R2\mathbb{R}^3 \to \mathbb{R}^2 defined by T(x, y, z) = (x + y, y - z) surjective
    • range(T) = R2\mathbb{R}^2 (entire codomain)
  • Projection transformation P: R3→R2\mathbb{R}^3 \to \mathbb{R}^2 defined by P(x, y, z) = (x, y) surjective
    • range(P) = R2\mathbb{R}^2 (entire codomain)
  • Linear transformation T: R2→R3\mathbb{R}^2 \to \mathbb{R}^3 defined by T(x, y) = (x, y, x+y) not surjective
    • range(T) = {(a, b, c) ∈ R3\mathbb{R}^3 | c = a + b} (proper subspace of R3\mathbb{R}^3)

Key Terms to Review (22)

Differential Operator: A differential operator is a mathematical operator defined as a function of the differentiation operator, which takes a function and produces another function by performing differentiation. In the context of linear transformations, it can be used to analyze the kernel and range by identifying how functions behave under differentiation, thus revealing important properties about the transformations themselves.
Dim(ker(t)): The term dim(ker(t)) refers to the dimension of the kernel of a linear transformation 't', which is the set of all vectors that are mapped to the zero vector by 't'. This dimension provides important information about the structure of the linear transformation, indicating how many dimensions of input vectors lead to a loss of information when transformed. Understanding dim(ker(t)) helps in grasping concepts like injectivity and the relationship between the kernel and the range of a transformation.
Dim(range(t)): The term dim(range(t)) refers to the dimension of the range of a linear transformation t, which is the number of linearly independent vectors that can be produced by the transformation from a given vector space. This dimension gives insight into how much of the target space can be reached through linear combinations of the images of vectors in the domain, effectively measuring the 'spread' or 'coverage' of the transformation. Understanding this dimension is crucial for analyzing properties such as injectivity and surjectivity in linear mappings.
Dim(v): The term dim(v) refers to the dimension of a vector space 'v', which is defined as the maximum number of linearly independent vectors in that space. Dimension provides insight into the structure of the vector space, indicating how many directions or degrees of freedom exist within it. In the context of linear transformations, understanding the dimension helps in analyzing properties like injectivity and surjectivity, as well as the relationship between the kernel and range.
Fundamental Theorem of Linear Algebra: The Fundamental Theorem of Linear Algebra describes the relationships between the four fundamental subspaces associated with a matrix: the column space, the row space, the null space, and the left null space. This theorem highlights the dimensions of these subspaces and establishes connections between the rank and nullity of a matrix, as well as its implications for solutions to linear equations and linear transformations.
Homogeneous Equation: A homogeneous equation is a type of linear equation that can be expressed in the form $Ax = 0$, where $A$ is a matrix and $x$ is a vector of variables. The key characteristic of a homogeneous equation is that it always includes the zero vector as a solution, making it fundamental to understanding the kernel of linear transformations. This zero solution indicates that if there are any non-trivial solutions, they will form a vector space, leading to deeper insights about the structure of the solutions.
Image: The image of a linear transformation is the set of all output vectors that can be produced by applying the transformation to the input vectors from the domain. It represents the range of the transformation and is crucial for understanding how transformations map elements from one vector space to another. The concept of image is linked to the kernel, as both are essential for characterizing the properties of linear transformations, particularly in terms of their injectivity and surjectivity.
Injective Linear Transformation: An injective linear transformation is a function between two vector spaces that maps distinct elements in the domain to distinct elements in the codomain. This means that if two different inputs produce the same output, then the transformation is not injective. The significance of injectivity relates closely to the concepts of kernel and range, as it indicates that the kernel only contains the zero vector, leading to a one-to-one correspondence between the input and output spaces.
Kernel: The kernel of a linear transformation is the set of all vectors that are mapped to the zero vector. This concept is essential in understanding the behavior of linear transformations, particularly regarding their injectivity and the relationship between different vector spaces. The kernel also plays a crucial role in determining properties like the rank-nullity theorem, which relates the dimensions of the kernel and range.
Linear Dependence: Linear dependence refers to a situation where a set of vectors in a vector space can be expressed as a linear combination of other vectors in the set. This means that at least one of the vectors can be represented as a sum of scalar multiples of the others, indicating that they do not provide independent directions in the space. Understanding linear dependence is crucial for identifying bases, span, and transformations in linear algebra, particularly when analyzing properties of vector spaces and linear transformations.
Linear map: A linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that for any vectors \( u \) and \( v \) in the first vector space and any scalar \( c \), a linear map \( T \) satisfies the properties \( T(u + v) = T(u) + T(v) \) and \( T(cu) = cT(u) \). Understanding linear maps is crucial as they relate directly to the concepts of kernel and range, which describe the output behavior of the map and its structure.
Linear operator: A linear operator is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if you take two vectors and add them, the operator will give you the same result as applying it to each vector first and then adding the results. Linear operators are central to understanding concepts like eigenvalues and eigenvectors, the kernel and range of transformations, invertibility, and can also be represented in Jordan canonical form, which simplifies their study and application.
Linear transformation: A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means if you take any two vectors and apply the transformation, the result will be the same as transforming each vector first and then adding them together. It connects to various concepts, showing how different bases interact, how they can change with respect to matrices, and how they impact the underlying structure of vector spaces.
Matrix transformation: A matrix transformation is a function that maps vectors from one vector space to another using a matrix. It provides a systematic way to apply linear transformations, allowing for operations such as rotation, scaling, and reflection in multidimensional spaces. This concept is crucial for understanding how linear mappings can be represented and manipulated using matrices.
Null Space: The null space of a matrix or a linear transformation is the set of all vectors that, when multiplied by that matrix or transformation, yield the zero vector. This concept is crucial in understanding the behavior of linear systems and provides insight into properties like linear independence, rank, and dimensions, as well as how solutions to linear equations can be interpreted geometrically as subspaces.
Nullity: Nullity refers to the dimension of the kernel of a linear transformation, which is the set of all vectors that are mapped to the zero vector. It is a measure of how many dimensions are 'lost' when the transformation is applied, giving insight into the structure of the transformation and its properties, such as whether it is injective or surjective. Understanding nullity helps in analyzing the relationship between input and output spaces in linear algebra.
Range: The range of a linear transformation is the set of all possible output vectors that can be produced by applying the transformation to every vector in the input space. This concept is crucial because it helps us understand how linear transformations can change or map spaces, indicating the dimensions of the output and providing insight into whether every output vector can be achieved from some input vector.
Rank: Rank is a fundamental concept in linear algebra that represents the maximum number of linearly independent column vectors in a matrix. It provides insights into the dimensions of the column space and row space, revealing important information about the solutions of linear systems, the behavior of linear transformations, and the structure of associated tensors.
Rank-Nullity Theorem: The Rank-Nullity Theorem states that for any linear transformation from one vector space to another, the sum of the rank (the dimension of the image) and the nullity (the dimension of the kernel) is equal to the dimension of the domain. This theorem helps illustrate relationships between different aspects of vector spaces and linear transformations, linking concepts like subspaces, linear independence, and matrix representations.
Surjective Linear Transformation: A surjective linear transformation is a type of function between two vector spaces that maps every element of the target space to at least one element from the source space. This means that the image of the transformation covers the entire target space, indicating that it is 'onto.' Surjective transformations are essential in understanding the range of a linear transformation, as they help define whether every possible output in the codomain can be achieved through some input from the domain.
Surjective Mapping: A surjective mapping, also known as an onto mapping, is a type of function where every element in the codomain has at least one corresponding element in the domain. This means that the mapping covers the entire codomain, ensuring that no elements are left out. Understanding surjectivity is important when examining linear transformations, as it connects directly to the concepts of range and kernel, shedding light on the nature of solutions to linear equations.
T: v → w: The notation 't: v → w' represents a linear transformation 't' that maps vectors from vector space 'v' to vector space 'w'. This mapping preserves the operations of vector addition and scalar multiplication, which are fundamental characteristics of linear transformations. Understanding this mapping is essential as it lays the foundation for examining the kernel and range of transformations, how they can be represented using matrices, and the conditions under which such transformations are invertible.
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