# Tensor Calculus Lecture 12b: Inner Products in Tensor Notation

TLDRThe video script delves into the concept of inner products within the framework of linear algebra, using tensor notation. It explains how to calculate the dot product of vectors represented by their components, emphasizing the role of the metric tensor, also known as the ground matrix. The script introduces index juggling to simplify notation and illustrates the relationship between different matrices, including the Gram matrix and the Jacobian matrix, highlighting the importance of matrix order and transposes. The discussion aims to clarify why certain matrices, despite representing symmetric transformations, may not be symmetric themselves.

###### Takeaways

- π The script discusses the concept of the inner product in the context of linear algebra and tensor notation.
- π§ It addresses the confusion about why a matrix representing a symmetric transformation is not symmetric itself, which will be explained later.
- π The script introduces the metric tensor, also known as the ground matrix, which represents the inner products of basis vectors.
- π The process of finding the dot product of vectors 'u' and 'v' using their components is explained, highlighting the tensor notation approach.
- π The script mentions the importance of index juggling in linear algebra, which allows for more compact and powerful notation.
- π The Gram matrix, or the metric tensor, is calculated for a specific basis, demonstrating the matrix form of the inner products of the basis vectors.
- π The script explains the relationship between the new Gram matrix and the original one, emphasizing the role of the Jacobian matrix in this transformation.
- π’ The importance of matrix order and transposes in matrix notation is highlighted, especially in the context of tensor notation.
- π The script illustrates how tensor notation dictates the use of matrices and the placement of indices, making it easier to remember the correct matrices to use.
- π€ The script promises to eventually explain why the matrix representing the metric tensor is not symmetric, despite the transformation being symmetric.
- π The final takeaway is the demonstration of how the new Gram matrix is related to the original one through matrix multiplication, involving the Jacobian matrix and its transpose.

###### Q & A

### What is the main topic discussed in the script?

-The main topic discussed in the script is the concept of inner products in the context of linear algebra, specifically focusing on the tensor notation and the metric tensor.

### Why is the matrix representing the inner product also called the metric tensor?

-The matrix representing the inner product is called the metric tensor because it contains the pairwise dot products of the basis vectors, which define the 'metric' or the way distances and angles are measured in the vector space.

### What is the significance of the term 'doubly covariant tensor' mentioned in the script?

-A 'doubly covariant tensor' refers to a tensor that has two lower indices, which means it transforms in a specific way under a change of basis, similar to how a vector transforms. This term is significant because it helps in understanding how the metric tensor relates to the change of basis.

### What is the difference between the 'gram matrix' and the 'metric tensor'?

-The 'gram matrix' is a specific instance of the 'metric tensor' where the basis vectors are orthonormal. The metric tensor, on the other hand, can be used for any basis, not just orthonormal ones.

### Why does the script mention 'index juggling' in the context of linear algebra?

-The script mentions 'index juggling' as a way to introduce a more compact and powerful notation in linear algebra, allowing for the manipulation of indices to express various mathematical operations in a concise manner.

### What is the purpose of the 'Jacobian relationship' in the context of the metric tensor?

-The Jacobian relationship is used to relate the metric tensors of different coordinate systems or bases. It shows how the components of the metric tensor change under a transformation from one basis to another.

### How does the script explain the process of finding the dot product of two vectors using their components?

-The script explains that the dot product of two vectors can be found by expressing each vector in terms of its basis components and then using the metric tensor to sum the products of the corresponding components.

### What is the importance of the order and transposes in matrix notation when converting tensor expressions to matrix form?

-The order and transposes in matrix notation are important because they ensure the correct contraction of indices in tensor expressions. This is crucial for maintaining the mathematical integrity of the operations when transitioning from tensor notation to matrix form.

### Why is the script discussing the symmetry of the matrix representing the inner product?

-The script discusses the symmetry of the matrix to address a conundrum where a symmetric transformation does not result in a symmetric matrix. This is an important concept to understand the properties of the metric tensor and its relation to the underlying linear transformations.

### What is the role of the basis vectors in the calculation of the inner product using tensor notation?

-The basis vectors play a crucial role in the calculation of the inner product using tensor notation. They define the 'ground matrix' or metric tensor, which is used to compute the inner product of any two vectors expressed in terms of these basis vectors.

### How does the script handle the transition from tensor notation to matrix form in the context of the inner product?

-The script handles the transition by first expressing the inner product in terms of the metric tensor using tensor notation, then showing how to rearrange and transpose the elements to form a matrix that can be used for calculations in linear algebra.

###### Outlines

##### π Introduction to Inner Product in Tensor Notation

This paragraph introduces the concept of the inner product within the context of linear algebra, specifically using tensor notation. The script discusses the confusion around the non-symmetric nature of a matrix representing a symmetric transformation, which will be revisited later. The focus then shifts to calculating the inner product (u dot v) using components, emphasizing that this is not limited to geometric vectors. The metric tensor, which represents the inner product, is introduced as the matrix of pairwise dot products of the basis vectors. The paragraph also touches on index juggling, a powerful notation technique in linear algebra, and ends with the calculation of the Gram matrix for a given basis, highlighting the importance of order and transposes in matrix notation.

##### π Matrix Representation and Symmetry of Inner Product

This paragraph delves into the matrix representation of the inner product, emphasizing the relationship between different matrices and the importance of tensor notation in dictating which matrices to use. It explains how the new Gram matrix is related to the original one through a Jacobian relationship, facilitated by the tensor's doubly covariant nature. The script also clarifies the need to transpose a matrix to achieve proper matrix multiplication due to index contraction. The paragraph concludes with a promise to explain why the matrix representing the inner product is not symmetric, setting the stage for a deeper exploration of this concept.

###### Mindmap

###### Keywords

##### π‘Inner Product

##### π‘Tensor Notation

##### π‘Symmetric Matrix

##### π‘Linear Transformation

##### π‘Metric Tensor

##### π‘Index Juggling

##### π‘Gram Matrix

##### π‘Basis Vectors

##### π‘Doubly Covariant Tensor

##### π‘Jacobian Matrix

###### Highlights

Introduction to the concept of the inner product in the context of tensor notation.

Explanation of why a matrix representing a symmetric transformation may not be symmetric itself.

The matrix representing the inner product is identified as the metric tensor.

Demonstration of how to determine the dot product of vectors using their components.

Use of tensor notation to generalize the dot product beyond geometric vectors.

The matrix of pairwise dot products of basis vectors is called the metric tensor.

Introduction of index juggling in linear algebra for more compact notation.

Calculation of the Gram matrix for a specific basis using tensor notation.

The Gram matrix is shown to be the metric tensor in matrix form.

Importance of matrix order and transposes in matrix notation.

The relationship between the new Gram matrix and the old one is explored.

Tensor notation is used to dictate the matrices involved in the relationship.

The concept of doubly covariant tensors is introduced.

Jacobian relationship is used to relate the new metric tensor to the old one.

Matrix representation of the Jacobian and its transpose is discussed.

Explanation of how tensor notation helps in remembering which matrices to use.

The final explanation of why the metric tensor matrix is not symmetric.

###### Transcripts

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