Tensors for Beginners 8: Linear Map Transformation Rules

eigenchris
25 Jan 201811:35
EducationalLearning
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TLDRThis video script explores the transformation rules of linear maps between different bases in a mathematical context. It explains how to determine the components of a vector transformed by a linear map, using matrix multiplication in the 'e' basis. The script delves into deriving a new matrix for a transformed basis, emphasizing the contravariant nature of vector components and the covariant transformation of basis vectors. It simplifies tensor notation using Einstein's summation convention, which omits explicit summation signs for clarity. The video concludes with an introduction to (1,0), (0,1), and (1,1)-tensors, setting the stage for the discussion of the metric tensor in a subsequent video.

Takeaways
  • πŸ“š The video discusses transformation rules for linear maps when transitioning between different bases.
  • πŸ” It begins with an example of a linear map represented by a matrix in the 'e' basis, which transforms basis vectors e_1 and e_2 in specific ways.
  • πŸ“ The script explains how to find the components of the vector L(V) by multiplying the column vector representing V by the matrix L, resulting in components [1/2, 2].
  • πŸ”„ The concept of contravariance is introduced, explaining the need for a backward transform when dealing with vector components changing bases.
  • πŸ†• The necessity to find a new matrix for the transformed basis is highlighted, involving determining L~ coefficients for the output vectors using the new basis vectors.
  • πŸ”’ A step-by-step process is outlined to find the L~ coefficients, involving rewriting new basis vectors in terms of the old, using linearity, and applying definitions and transformations.
  • 🧩 The video simplifies the process by illustrating how to transform matrix coordinates from one basis to another using both forward and backward transformations.
  • πŸ“‰ It demonstrates the practical application of the new matrix L~ in transforming the basis vector components and verifies the results using the new basis.
  • πŸŽ“ The script introduces Einstein's notation as a shorthand for tensor components, simplifying the notation by omitting summation signs when indices are repeated.
  • πŸ“š The video concludes with the classification of different types of tensors: (1,0)-tensors for basis covectors and vector components, (0,1)-tensors for basis vectors and covector components, and (1,1)-tensors for linear maps.
  • πŸ”š The final takeaway is theι’„ε‘Šof the next video, which will cover the metric tensor as the final example of a tensor.
Q & A
  • What are the transformation rules for linear maps when changing from one basis to another?

    -The transformation rules for linear maps involve using both the forward and backward transformations. To find the new matrix representation in a different basis, you multiply the original matrix by the backward transformation matrix on the left and by the forward transformation matrix on the right.

  • Why does the linear map turn e_1 into 0.5 e_1 and e_2 into 2 e_2?

    -The linear map's transformation of the basis vectors e_1 and e_2 is defined by the matrix that represents the linear map in the given 'e' basis. The transformation is a result of the matrix's effect on the basis vectors, scaling them according to the matrix's elements.

  • What is the significance of the vector V with components [1,1] in the script?

    -Vector V with components [1,1] is used as an example to demonstrate how the linear map L transforms a vector when the transformation is represented in the 'e' basis. The transformation results in the new vector components [1/2, 2].

  • How do you determine the components of the output vector L(V) in the 'e' basis?

    -To determine the components of L(V), you multiply the column vector representing V by the matrix representing the linear map L. This multiplication yields the new components of the transformed vector in the 'e' basis.

  • What is the contravariant transformation of vector components?

    -The contravariant transformation of vector components involves applying the backward transformation to the components in the old basis to obtain the components in the new basis. This is because vector components are contravariant, meaning they transform in the opposite way to the basis vectors.

  • What does the L~ matrix represent in the context of the script?

    -The L~ matrix represents the new matrix that describes how to construct output vectors using the new basis vectors (e~). It is derived by applying the transformation rules for linear maps to the original matrix L.

  • How can you simplify the process of transforming matrix components using Einstein's notation?

    -Einstein's notation simplifies the process by omitting summation signs when an index variable appears on both the top and bottom of a term. This implies that a summation over that index is understood to be present, making the notation more concise and easier to work with.

  • What is the role of the Kronecker delta in transforming matrix components?

    -The Kronecker delta acts as a summation index canceller. When multiplying a matrix by the identity matrix, the Kronecker delta allows for the cancellation of summation indices, effectively leaving the original matrix components unchanged.

  • How does the script define (1,0)-tensors, (0,1)-tensors, and (1,1)-tensors?

    -In the script, (1,0)-tensors are basis covectors and vector components that transform using the contravariant law. (0,1)-tensors are basis vectors and covector components that transform using the covariant law. (1,1)-tensors are linear maps that transform using both the forward and backward matrices, making them both contravariant and covariant.

  • What is the final example of a tensor that the script mentions for the next video?

    -The script mentions that the final example of a tensor to be discussed in the next video is the metric tensor.

Outlines
00:00
πŸ“š Linear Map Transformation Rules

This paragraph introduces the concept of how linear maps transform when switching between different bases. It uses a matrix representation of a linear map 'L' expressed in the 'e' basis, which turns e_1 into 0.5 e_1 and e_2 into 2 e_2. The process of finding the components of the vector L(V) with components [1,1] is demonstrated by matrix multiplication, resulting in [1/2, 2]. The paragraph also discusses the backward transformation and the need to find new matrix coefficients for the transformed basis vectors, emphasizing the contravariant nature of vector components.

05:05
πŸ” Deriving New Basis Matrix for Linear Maps

The focus of this paragraph is on deriving the new matrix representation for a linear map in a different basis. It explains the process of using the forward and backward transformations to find the new matrix 'L~' that operates in the new basis. The paragraph illustrates the steps to transform the basis vectors and solve for the 'L~' coefficients. It also simplifies the derivation by dropping summation signs and using Einstein's notation, which assumes summation over repeated indices. The result is a more concise and clear derivation, leading to the final transformation of the matrix components using both forward and backward transformations.

10:06
🎯 Simplifying Tensor Transformations with Einstein's Notation

This paragraph discusses the simplification of tensor transformations using Einstein's notation, which omits explicit summation signs for repeated indices. It explains how this notation makes derivations more concise and clear. The paragraph also introduces the concept of Kronecker deltas as summation index cancellers and demonstrates their use in simplifying matrix multiplication with the identity matrix. The paragraph concludes with a quick derivation of the backward transformation for matrix components using the new notation and the properties of Kronecker deltas, showcasing the efficiency of Einstein's notation in tensor calculus.

🧭 Tensor Classification and Transformation Laws

The final paragraph summarizes the key concepts of tensor transformations and introduces the classification of tensors based on their transformation laws. It explains that basis covector components and vector components, which are contravariant, are referred to as (1,0)-tensors. Basis vectors and covector components, which are covariant, are called (0,1)-tensors. Linear maps, which transform using both forward and backward transformations, are classified as (1,1)-tensors. The paragraph also previews the next topic, the metric tensor, which will be the final example of a tensor discussed in the subsequent video.

Mindmap
Keywords
πŸ’‘Linear Map
A linear map, also known as a linear transformation, is a function that maps vectors from one vector space to another while preserving the operations of vector addition and scalar multiplication. In the video, the linear map is represented by a matrix and is used to transform basis vectors, illustrating how it turns e_1 into 0.5 e_1 and e_2 into 2 e_2, which is fundamental to understanding transformations between different bases.
πŸ’‘Basis
A basis is a set of linearly independent vectors that span a vector space. The script discusses how a linear map is initially expressed in the 'e' basis, meaning that the transformations are initially described using a specific set of basis vectors. The concept is crucial as it sets the stage for discussing transformations between different bases.
πŸ’‘Matrix
In linear algebra, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent linear maps. The video uses a matrix to represent the linear map and shows how it is used to calculate the transformed components of a vector in a given basis.
πŸ’‘Vector Components
Vector components are the scalar values that describe the position of a vector in a given basis. The script explains how to find the components of the vector L(V) by multiplying the vector V by the matrix representing the linear map, yielding the components [1/2, 2] in the 'e' basis.
πŸ’‘Backward Transform
A backward transform is used to convert vector components from one basis to another. The video script mentions applying the backward transform to obtain components in a new basis, which is essential for understanding how vector representations change with the choice of basis.
πŸ’‘Contravariant
In the context of tensor analysis, contravariant vectors change components in the opposite way to the change in the basis. The script explains that vector components are contravariant, which is why the backward transform is used to find components in a new basis.
πŸ’‘(1,0)-Tensor
A (1,0)-tensor, as introduced in the script, refers to a tensor that transforms with the contravariant law, which includes basis covectors and vector components. The concept is part of a broader discussion on how different types of tensors transform under changes of basis.
πŸ’‘Forward Transform
The forward transform is the process of expressing new basis vectors in terms of the old basis vectors. In the video, the forward transform is used to rewrite the new basis vectors in terms of the old ones, which is a step in finding the new matrix that represents the linear map in the new basis.
πŸ’‘Einstein's Notation
Einstein's notation, also known as the Einstein summation convention, is a shorthand notation that omits the summation signs in expressions involving multiple summations over the same index. The script introduces this notation as a way to simplify the tensor transformation derivations, making the mathematical expressions more concise and easier to handle.
πŸ’‘Metric Tensor
The metric tensor is a type of tensor that defines the geometry of a space, including how distances and angles are measured. While not the main focus of the script, the video mentions that the metric tensor will be the subject of the next video, indicating a continuation of the discussion on tensors and their applications.
Highlights

Exploration of transformation rules for linear maps when transitioning between bases.

Introduction of a linear map represented by a matrix in the 'e' basis.

Explanation of how the linear map transforms basis vectors e_1 and e_2.

Multiplication of a vector's column vector by the matrix to find the transformed vector's components.

Discussion on the contravariant nature of vector components and the use of the backward transform.

The need to find a new matrix for the e~ basis vectors to determine output vector components.

Derivation of the new matrix coefficients L~ using the forward and backward transformations.

Use of linearity of L to simplify the expression for the transformed basis vectors.

Rearrangement of sums and rewriting of basis vectors to solve for L~ coefficients.

Transformation of matrix coordinates from old to new basis by multiplying with backward and forward transforms.

Practical demonstration of transforming a matrix L~ in the new basis using multiplication.

Verification of the new L~ matrix by transforming new basis vector components.

Introduction of Einstein's notation for simplifying tensor component expressions.

Derivation of the backward transformation for matrix components using Einstein's notation.

Explanation of Kronecker delta as a summation index canceller in tensor expressions.

Categorization of tensors based on their transformation properties: (1,0), (0,1), and (1,1)-tensors.

Upcoming introduction of the metric tensor as the final example of a tensor in the next video.

Transcripts
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