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The matrix of a linear transformation

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Intros
Lessons
  1. The Matrix of a Linear Transformation Overview:
  2. The Standard Basis and Matrix
    • T(x)=Ax,AT(x)=Ax, A: The Standard Matrix
    • R2\Bbb{R}^2 standard Basis: r^2 standard basis e1 and r^2 standard basis e2
    • R3\Bbb{R}^3 standard Basis: r^3 standard basis e1, e2, e3
    • Transformed standard basis
    • Finding the Standard Matrix
  3. Finding the Standard Matrix Geometrically in R2\Bbb{R}^2
    • Drawing the Standard basis on a Graph
    • Identifying the Transformed Standard Basis
    • Combining the Standard Basis' into a Matrix
  4. Types of Geometric Linear Transformations in R2\Bbb{R}^2
    • Reflections on a Line
    • Horizontal Expansion/Contraction
    • Vertical Expansion/Contraction
    • Horizontal & Vertical Shears
    • Projections
    • Rotation Transformations
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Examples
Lessons
  1. Finding the Standard Matrix with Transformed Basis
    Assume that TT is a linear transformation. Find the standard matrix of TT if standard matrix of T, T(e1) standard matrix of T, T(e2) where standard matrix of T, e1= [1 0] and standard matrix of T, e2= [0 1].
    1. Assume that TT is a linear transformation. Find the standard matrix of TT if standard matrix of T, T(e1) standard matrix of T, T(e2), T(e3) where standard matrix of T, e1= [1 0 0], e2= [0 1 0] and standard matrix of T, e3= [0 0 1].
      1. Finding the Standard Matrix Geometrically
        Assume that TT is a linear transformation. Find the standard matrix of TT if T:R2T: \Bbb{R}^2 →R2 \Bbb{R}^2 is a vertical shear transformation that maps e1e_1 to e1+2e2e_1+2e_2 and leaves e2e_2 unchanged.
        1. Assume that TT is a linear transformation. Find the standard matrix of TT if T:R2T: \Bbb{R}^2 →R2 \Bbb{R}^2 first performs a x2=x1x_2=x_1 reflection, and then a x1x_1 reflection.
          1. Finding the Matrix Algebraically
            Show that TT is a linear transformation by finding a matrix AA that implements the mapping:
            Finding the Matrix Algebraically and show linear transformation
            1. Finding for x given the Transformation
              Let T:R2T: \Bbb{R}^2 →R2 \Bbb{R}^2 be a linear transformation such that Finding for x given the Transformation. Find a xx such that Finding for x given the Transformation
              Topic Notes
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              Introduction to the Matrix of a Linear Transformation

              Welcome to our exploration of the matrix of a linear transformation, a fundamental concept in linear algebra! This topic is crucial for understanding how linear transformations work and how we can represent them mathematically. The matrix of a linear transformation, also known as the standard matrix, is a powerful tool that allows us to describe and analyze linear transformations in a concise, numerical format. In our introduction video, we'll dive into this concept, showing you how to construct and interpret these matrices. You'll see how they relate to the input and output vectors of a transformation, and why they're so useful in various mathematical and real-world applications. Understanding this concept will significantly enhance your grasp of linear algebra and its practical uses. So, let's embark on this exciting journey together, unraveling the mysteries of linear transformations through their matrix representations!

              Understanding the Standard Matrix

              Definition of the Standard Matrix

              The standard matrix is a fundamental concept in linear algebra applications that represents a linear transformation in matrix form. It is a powerful tool used to describe how vectors are transformed from one space to another. Essentially, a standard matrix encapsulates the entire linear transformation in a single, compact form.

              Significance in Linear Transformations

              The importance of the standard matrix in linear transformations cannot be overstated. It provides a systematic way to apply transformations to vectors and understand their effects. By using the standard matrix, complex transformations can be simplified into matrix multiplication operations, making calculations more efficient and intuitive.

              Constructing the Standard Matrix

              To construct a standard matrix for a given linear transformation, we need to determine how the transformation affects the standard basis vectors. In a two-dimensional space, these are typically represented as e1 = (1,0) and e2 = (0,1). The transformed versions of these vectors become the columns of the standard matrix. This process effectively captures the essence of the transformation in matrix form.

              Examples of Standard Matrices

              Let's consider some examples to illustrate the concept of standard matrices:

              1. Rotation: A 90-degree counterclockwise rotation in 2D space can be represented by the standard matrix [[0, -1], [1, 0]].
              2. Scaling: A transformation that doubles all coordinates is represented by the matrix [[2, 0], [0, 2]].
              3. Reflection: Reflection across the y-axis is given by the matrix [[-1, 0], [0, 1]].

              Applications of Standard Matrices

              Standard matrices find applications in various fields, including computer graphics, physics, and engineering. In computer graphics, for instance, transformation matrices are used to rotate, scale, and translate objects in 2D and 3D spaces. In physics, they help describe the motion and orientation of objects under different forces.

              Properties of Standard Matrices

              Standard matrices possess several important properties that make them invaluable in linear algebra:

              • Linearity: They preserve vector addition and scalar multiplication.
              • Composition: Multiple transformations can be combined by multiplying their respective standard matrices.
              • Invertibility: If a standard matrix is invertible, it represents a reversible transformation.

              Relationship to Other Concepts

              The standard matrix is closely related to other key concepts in linear algebra. It forms the basis for understanding eigenvalues and eigenvectors, which are crucial in analyzing the long-term behavior of transformations. Additionally, the determinant of a standard matrix provides insights into how the transformation affects area or volume in the transformed space.

              Challenges and Considerations

              While standard matrices are powerful tools, it's important to note that not all transformations can be represented by matrices. Non-linear transformations, for example, require different mathematical approaches. Additionally, as the dimensionality of the space increases, working with standard matrices can become computationally intensive, necessitating more advanced techniques.

              Conclusion

              The standard matrix is a cornerstone concept in linear algebra applications, providing a concise and powerful way to represent linear transformations. By encapsulating complex transformations in matrix form, it simplifies calculations and offers insights into the nature of these transformations. Whether you're studying mathematics, working in computer graphics, or applying linear algebra in engineering, understanding standard matrices is crucial for mastering linear transformations and their wide-ranging applications.

              The Standard Basis and Its Role

              The standard basis is a fundamental concept in linear algebra that plays a crucial role in finding the standard matrix of a linear transformation. It consists of a set of vectors that form a basis for a vector space, with each vector having a 1 in one position and 0s in all others. Understanding the standard basis is essential for various mathematical operations and representations in different dimensions.

              In R2 (two-dimensional space), the standard basis consists of two vectors: e1 = (1,0) and e2 = (0,1). These vectors represent the unit vectors along the x and y axes, respectively. For example, any point in R2 can be expressed as a linear combination of these basis vectors. The point (3,4) can be written as 3e1 + 4e2.

              Moving to R3 (three-dimensional space), the standard basis expands to three vectors: e1 = (1,0,0), e2 = (0,1,0), and e3 = (0,0,1). These vectors correspond to the unit vectors along the x, y, and z axes. For instance, the point (2,3,-1) in R3 can be expressed as 2e1 + 3e2 - 1e3.

              The concept of the standard basis extends to higher dimensions in RN, where N represents the number of dimensions. In RN, the standard basis consists of N vectors, each with a 1 in a different position and 0s elsewhere. This allows for the representation of any point in N-dimensional space as a linear combination of these basis vectors.

              The importance of the standard basis becomes evident when finding the standard matrix of a linear transformation. A linear transformation T: RN RM can be completely described by its effect on the standard basis vectors. To find the standard matrix, we apply the transformation to each standard basis vector and use the resulting vectors as columns of the matrix.

              For example, consider a linear transformation T: R2 R2 defined by T(x,y) = (2x+y, x-y). To find the standard matrix, we apply T to the standard basis vectors of R2:

              T(e1) = T(1,0) = (2,1)

              T(e2) = T(0,1) = (1,-1)

              The standard matrix A is then formed by using these vectors as columns:

              A = [2 1]

              [1 -1]

              This process demonstrates how the standard basis serves as a bridge between the abstract concept of a linear transformation and its concrete matrix representation. By applying the transformation to the standard basis vectors, we capture all the necessary information to represent the transformation as a matrix.

              In conclusion, the standard basis is a powerful tool in linear algebra, providing a consistent framework for representing vectors and transformations across different dimensions. Its simplicity and universality make it an indispensable concept in various mathematical and practical applications, from computer graphics to data analysis. Understanding and utilizing the standard basis is key to mastering linear transformations and their matrix representations.

              Steps to Find the Standard Matrix

              Finding the standard matrix of a linear transformation is a crucial skill in linear algebra. This step-by-step guide will walk you through the process, using the example from the video to illustrate each step. Understanding how to find the standard matrix is essential for analyzing and applying linear transformations in various mathematical and real-world contexts.

              Step 1: Identify the Domain and Codomain

              First, determine the vector spaces involved in the transformation. In our example, we're working with a transformation T from R² to R². This means both the input and output vectors have two components.

              Step 2: Apply the Transformation to Basis Vectors

              The key to finding the standard matrix is to apply the transformation to the standard basis vectors of the domain. For R², these are e = (1,0) and e = (0,1). In our example:

              • T(e) = T(1,0) = (1,1)
              • T(e) = T(0,1) = (-1,2)

              This step is crucial because it reveals how the transformation affects the fundamental building blocks of the vector space.

              Step 3: Construct the Standard Matrix

              The standard matrix A is formed by using the transformed basis vectors as columns. In our case:

              A = [1 -1] [1 2]

              This step is important because it encapsulates the entire linear transformation in a single matrix, allowing for easy computation and analysis.

              Step 4: Verify the Matrix

              To ensure correctness, verify that Ax = T(x) for any vector x. For example, let x = (2,3). Then:

              Ax = [1 -1] [2] = [2-3] = [-1] [1 2] [3] [2+6] [ 8]

              This should match the result of applying T directly to (2,3).

              Understanding the Process

              The standard matrix is powerful because it allows us to represent the linear transformation as matrix multiplication. This is based on the fundamental property that linear transformations preserve linear combinations. By determining how the transformation affects the basis vectors, we capture its essence for all vectors in the space.

              Why This Method Works

              This method works because any vector in R² can be expressed as a linear combination of the standard basis vectors. For example, (a,b) = a(1,0) + b(0,1). The linearity of the transformation means that:

              T(a,b) = T(a(1,0) + b(0,1)) = aT(1,0) + bT(0,1)

              This is exactly what matrix multiplication with our standard matrix A accomplishes.

              Applications and Importance

              Understanding how to find the standard matrix is crucial for various applications in linear algebra, including:

              • Analyzing geometric transformations like rotations, reflections, and scaling
              • Solving systems of linear equations
              • Studying eigenvalues and eigenvectors
              • Applying linear transformations in computer graphics and image processing

              Conclusion

              Finding the standard matrix of a linear transformation is a fundamental skill in linear algebra. By following these steps - identifying the domain and codomain, applying the transformation to basis vectors, constructing the matrix, and verifying the result - you can effectively represent any linear transformation as a matrix. This representation simplifies computations and provides deep insights into the nature of the transformation. Practice with various examples to solidify your understanding of this essential concept.

              Types of Linear Transformations and Their Standard Matrices

              Linear transformations are fundamental concepts in linear algebra that map vectors from one space to another while preserving vector addition and scalar multiplication. In this section, we'll explore various types of linear transformations and how to find their standard matrices, providing geometric interpretations for each.

              1. Reflections

              Reflections are transformations that flip vectors across a line or plane. For example, a reflection across the y-axis in 2D space can be represented by the matrix [[-1, 0], [0, 1]]. This transformation changes the sign of the x-coordinate while leaving the y-coordinate unchanged. Geometrically, it creates a mirror image of the original vector or shape.

              2. Expansions and Contractions

              Expansions and contractions scale vectors by a factor. An expansion increases the magnitude of vectors, while a contraction decreases it. The standard matrix for a uniform scaling in 2D is [[k, 0], [0, k]], where k > 1 for expansion and 0 < k < 1 for contraction. Geometrically, these transformations enlarge or shrink objects while maintaining their shape and orientation.

              3. Shears

              Shear transformations distort shapes by shifting one part of the object more than another. In 2D, a horizontal shear can be represented by the matrix [[1, k], [0, 1]], where k is the shear factor. This transformation slants objects, keeping one axis fixed while shifting points along the other axis proportionally to their distance from the fixed axis.

              4. Projections

              Projections map vectors onto a line or plane. For instance, a projection onto the x-axis in 2D has the matrix [[1, 0], [0, 0]]. This transformation collapses all points onto the x-axis, effectively reducing the dimension of the space. Geometrically, projections can be visualized as shadows cast by objects onto a line or plane.

              5. Rotations

              Rotations turn vectors around a fixed point or axis. In 2D, a counterclockwise rotation by angle θ is represented by the matrix [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]. Rotations preserve the length of vectors and the angles between them, resulting in a circular motion around the origin.

              Finding Standard Matrices

              To find the standard matrix for a linear transformation, we can apply the transformation to the standard basis vectors and use the resulting vectors as columns of the matrix. For example, in 2D:

              • e1 = [1, 0] and e2 = [0, 1] are the standard basis vectors
              • Apply the transformation T to e1 and e2
              • The standard matrix A is then [T(e1) | T(e2)]

              Geometric Interpretations

              Each type of linear transformation has a unique geometric interpretation:

              • Reflections: Mirror images across a line or plane
              • Expansions/Contractions: Uniform scaling of objects
              • Shears: Slanting or skewing of shapes
              • Projections: Collapsing objects onto a lower-dimensional space
              • Rotations: Circular motion around a fixed point or axis

              Understanding these geometric interpretations is crucial for visualizing how linear transformations affect vectors and shapes in space. It allows us to predict the outcome of transformations and design appropriate matrices for desired effects in various applications, such as computer graphics, physics simulations, and data analysis.

              Composition of Transformations

              Linear transformations can be combined through matrix multiplication to create more complex transformations. For instance, a

              Practical Applications and Examples

              Finding the standard matrix of a linear transformation is a crucial skill in linear algebra with numerous practical applications. Let's explore some examples and real-world scenarios to illustrate its importance.

              Example 1: Rotation in 2D Space

              Consider a linear transformation that rotates vectors in R² by 90° counterclockwise. To find the standard matrix, we need to determine how this transformation affects the standard basis vectors:

              T(1,0) = (0,1) and T(0,1) = (-1,0)

              The standard matrix A is then:

              A = [0 -1; 1 0]

              Example 2: Scaling in 3D Space

              Let's find the standard matrix for a transformation that doubles the x-coordinate, triples the y-coordinate, and leaves the z-coordinate unchanged:

              T(1,0,0) = (2,0,0), T(0,1,0) = (0,3,0), T(0,0,1) = (0,0,1)

              The standard matrix A is:

              A = [2 0 0; 0 3 0; 0 0 1]

              Example 3: Projection onto a Plane

              Consider a transformation that projects vectors onto the xy-plane in R³. This means the z-component becomes zero:

              T(1,0,0) = (1,0,0), T(0,1,0) = (0,1,0), T(0,0,1) = (0,0,0)

              The standard matrix A is:

              A = [1 0 0; 0 1 0; 0 0 0]

              Real-world applications of these transformations are abundant:

              1. Computer Graphics: Rotation matrices are used to manipulate 3D objects in video games and animation software.

              2. Image Processing: Scaling matrices can be applied to resize images or adjust their proportions.

              3. Data Compression: Projection matrices are used in techniques like Principal Component Analysis (PCA) for dimensionality reduction in large datasets.

              4. Physics simulations: Linear transformations model various physical phenomena, such as reflections, shears, and stretches.

              5. Machine Learning: Transformations are crucial in feature engineering and data preprocessing for many algorithms.

              Understanding how to find the standard matrix of a linear transformation is essential for more advanced concepts in linear algebra, such as eigenvalues and eigenvectors, which have applications in quantum mechanics, vibration analysis, and population dynamics.

              To solve problems involving standard matrices, follow these steps:

              1. Identify how the transformation affects each standard basis vector.

              2. Write these transformed vectors as columns of a matrix.

              3. The resulting matrix is the standard matrix of the transformation.

              Practice Problem: Find the standard matrix for a linear transformation T: R² R² that reflects vectors across the line y = x.

              Solution:

              1. T(1,0) = (0,1) (the x-axis is reflected to the y-axis)

              2. T(0,1) = (1,0) (the y-axis is reflected to the x-axis)

              3. The standard matrix A is therefore:

              A = [0 1; 1 0]

              By mastering the concept of standard matrices, you'll be well-equipped to tackle more complex problems in linear algebra and its applications across various fields of science, engineering, and data analysis, including eigenvalues and eigenvectors and physics simulations: linear transformations.

              Conclusion

              In this exploration of finding the standard matrix of a linear transformation, we've covered crucial concepts that form the foundation of linear algebra. The introduction video played a pivotal role in demystifying this complex topic, providing a clear roadmap for understanding. We learned that the standard matrix encapsulates the essence of a linear transformation, allowing us to represent it algebraically. Key points included the step-by-step process of determining the transformation's effect on basis vectors and constructing the matrix accordingly. It's essential to practice these techniques to solidify your understanding. As you continue your journey in linear algebra, remember that mastering the standard matrix of a linear transformation opens doors to more advanced concepts. We encourage you to explore further examples and applications. Don't hesitate to revisit the introduction video for reinforcement. Ready to deepen your knowledge? Engage with more problems and seek out additional resources to enhance your grasp of this fundamental concept in linear algebra.

              The Matrix of a Linear Transformation Overview:

              The Matrix of a Linear Transformation Overview: The Standard Basis and Matrix
              • T(x)=Ax,AT(x)=Ax, A: The Standard Matrix
              • R2\Bbb{R}^2 standard Basis: r^2 standard basis e1 and r^2 standard basis e2
              • R3\Bbb{R}^3 standard Basis: r^3 standard basis e1, e2, e3
              • Transformed standard basis
              • Finding the Standard Matrix

              Step 1: Understanding the Linear Transformation

              To begin with, we need to understand the function T(x)=AxT(x) = Ax. Here, TT represents the transformation, xx is the vector being transformed, and AA is the matrix that performs the transformation. The matrix AA is what we refer to as the standard matrix. In this context, the transformation takes a vector xx and produces a new vector by multiplying it with the matrix AA.

              Step 2: Introduction to the Standard Basis

              Before we can find the standard matrix, we need to understand the concept of the standard basis. The standard basis is a set of vectors that are used to define the space in which we are working. For example, in R2\Bbb{R}^2, the standard basis vectors are e1=(1,0)e_1 = (1, 0) and e2=(0,1)e_2 = (0, 1). In R3\Bbb{R}^3, the standard basis vectors are e1=(1,0,0)e_1 = (1, 0, 0), e2=(0,1,0)e_2 = (0, 1, 0), and e3=(0,0,1)e_3 = (0, 0, 1). These vectors are crucial because they help us understand how the transformation affects the space.

              Step 3: Applying the Transformation to the Standard Basis

              To find the standard matrix AA, we need to see how the transformation TT affects the standard basis vectors. For instance, if we are working in R2\Bbb{R}^2, we need to determine what T(e1)T(e_1) and T(e2)T(e_2) are. Suppose T(e1)=(2,4)T(e_1) = (2, 4) and T(e2)=(1,7)T(e_2) = (1, 7). These transformed vectors will form the columns of the matrix AA.

              Step 4: Constructing the Standard Matrix

              Once we have the transformed standard basis vectors, we can construct the standard matrix AA. The matrix AA is formed by placing the transformed vectors as columns in the matrix. For example, if T(e1)=(2,4)T(e_1) = (2, 4) and T(e2)=(1,7)T(e_2) = (1, 7), then the matrix AA will be: 

                  A = | 2  1 |
        | 4  7 |
              This matrix AA is the standard matrix for the transformation TT.

              Step 5: Verifying the Standard Matrix

              To ensure that the matrix AA is correct, we can verify it by applying it to the standard basis vectors and checking if we get the transformed vectors. For example, multiplying AA by e1e_1 should give us T(e1)T(e_1), and multiplying AA by e2e_2 should give us T(e2)T(e_2). If these conditions are met, then we have correctly found the standard matrix.

              Step 6: Generalizing to Higher Dimensions

              The process of finding the standard matrix can be generalized to higher dimensions. For Rn\Bbb{R}^n, we need to find the transformation of each standard basis vector eie_i where ii ranges from 1 to nn. The transformed vectors will form the columns of the matrix AA. This method ensures that we can find the standard matrix for any linear transformation in any dimension.

              Conclusion

              In summary, finding the standard matrix of a linear transformation involves understanding the transformation function, applying it to the standard basis vectors, constructing the matrix from the transformed vectors, and verifying the result. This process is fundamental in linear algebra and helps us understand how transformations affect different spaces.

              FAQs

              Here are some frequently asked questions about finding the standard matrix of a linear transformation:

              1. What is the standard matrix of a linear transformation?

                The standard matrix of a linear transformation is a matrix that represents the transformation in terms of its effect on the standard basis vectors. It allows us to apply the transformation to any vector through matrix multiplication.

              2. How do you find the matrix of a linear transformation?

                To find the matrix of a linear transformation, apply the transformation to each standard basis vector and use the resulting vectors as columns of the matrix. For example, in R², find T(1,0) and T(0,1), then use these as columns of the 2x2 matrix.

              3. What is the standard form of a linear transformation?

                The standard form of a linear transformation is its representation as a matrix. This matrix, when multiplied with any input vector, produces the same result as applying the transformation directly to that vector.

              4. Is the standard matrix of a linear transformation unique?

                Yes, the standard matrix of a linear transformation is unique for a given basis. However, the same transformation can have different matrix representations if different bases are used.

              5. How are linear transformations used in real-world applications?

                Linear transformations are used in various fields such as computer graphics (for rotations and scaling), data compression, image processing, and physics simulations. They provide a mathematical framework for manipulating vectors and spaces in many scientific and engineering applications.

              Prerequisite Topics

              Understanding the matrix of a linear transformation is a crucial concept in linear algebra, but to fully grasp its significance, it's essential to have a solid foundation in several prerequisite topics. These fundamental concepts provide the necessary context and tools to comprehend how matrices represent linear transformations.

              First and foremost, familiarizing yourself with the properties of linear transformation is vital. These properties form the backbone of how linear transformations behave and interact with vector spaces. Equally important are the properties of matrix multiplication, as they directly relate to how linear transformations can be composed and applied in sequence.

              The concept of eigenvalues and eigenvectors plays a crucial role in understanding the characteristics of linear transformations. These special vectors and their corresponding scalars provide insights into the transformation's behavior and its effect on the vector space. Additionally, grasping the idea of linear combination of vectors is essential, as it forms the basis for understanding how linear transformations act on vector spaces.

              Another key aspect is comprehending the image and range of linear transformations. This concept helps in visualizing the output of a transformation and understanding its effect on the entire vector space. Moreover, knowing the process of finding the transformation matrix is crucial, as it bridges the gap between the abstract concept of a linear transformation and its concrete representation as a matrix.

              Lastly, understanding linear equation applications provides practical context for how linear transformations and their matrices are used in real-world scenarios. This knowledge helps in appreciating the broader implications and uses of linear transformations beyond abstract mathematical concepts.

              By mastering these prerequisite topics, students will be well-equipped to tackle the complexities of matrices representing linear transformations. Each concept builds upon the others, creating a comprehensive understanding of how linear algebra describes and manipulates vector spaces through transformations. This foundational knowledge is not just academic; it's essential for various fields, including computer graphics, data analysis, and engineering applications where linear transformations play a pivotal role.

              The standard basis, e1e_1 and e2e_2, are unit vectors in R2\Bbb{R}^2 such that:
              Standard basis, unit vectors in r^2
              If transformed vectors are the transformed vectors, then the standard matrix is
              Standard matrix

              Why does that work? Watch the intro video

              The standard basis, e1e_1, e2e_2, and e3e_3 are unit vectors in R3\Bbb{R}^3 such that:
              unit vectors of standard basic e_1, e_2, e_3

              If transformed vectors e_1, e_2, e_3 are the transformed vectors, then the standard matrix is
              Standard matrix e_1, e_2, e_3

              To find the standard basis in R2\Bbb{R}^2 geometrically in a graph we:
              1. Draw the standard basis e1e_1 and e2e_2 in the x1x2x_1 x_2 plane
              2. Draw the transformed vectors using the information given
              3. Identify the transformed vectors T(e1)T(e_1), and T(e2)T(e_2).
              4. Combine them to get the standard matrix

              Here are the many types of transformations you may see in this section:

              Reflections
              x1-axis reflection x2-axis reflection
              x2=x1 reflection origin reflection

              Vertical/Horizontal Expansions and contractions
              horizontal expansion/contraction vertical expansion/contraction

              Vertical/Horizontal Shears
              horizontal shear vertical shear

              Projections
              projection onto x1 axis projection onto x2 axis

              Circle Rotation
              circle rotation transformation