# Tensors for Beginners 12: Bilinear Forms are Covector-Covector pairs

TLDRThis video introduces the concept of covector-covector pairs as bilinear forms, utilizing a non-standard tensor product notation for clarity. It discusses the benefits of viewing linear maps as vector-covector pairs, including simplified transformation rules and automatic derivation of matrix-vector multiplication formulas. The video also explores the perspective shift for bilinear forms, demonstrating how they can be represented as linear combinations of covector pairs, leading to transformation rules, multiplication formulas, and the correct array shape for bilinear operations.

###### Takeaways

- π The video introduces the concept of covector-covector pairs and their role in bilinear forms.
- π Non-standard notation for the tensor product is used, omitting the circle-times operator.
- π Linear maps can be represented as linear combinations of vector-covector pairs, simplifying transformation rules.
- π Basis vectors are covariant and transform using the backward transform B, while basis covectors are contravariant and transform using the forward transform F.
- π The benefits of the new perspective on linear maps include simplified transformation rules, automatic matrix-vector multiplication formulas, and correct array shapes for tensors.
- π Bilinear forms are viewed as linear combinations of covector-covector pairs, which aligns with their two-vector input nature.
- π Transformation rules for bilinear forms are derived by transforming the basis covectors individually using the forward transform F.
- π The component multiplication formula for bilinear forms acting on two vector inputs is obtained by applying linearity and index cancellation rules.
- π The array shape for bilinear forms is a row of rows, which makes sense when considering matrix multiplication with vectors written as columns.
- π§ The video emphasizes the practicality of viewing bilinear forms as arrays and the advantages this perspective brings to understanding transformation and multiplication rules.
- π Summary: Bilinear forms can be expressed as linear combinations of covector-covector pairs, providing transformation rules, multiplication formulas, and array shapes.

###### Q & A

### What is the main topic of the video?

-The main topic of the video is the introduction of covector-covector pairs and the demonstration that these pairs can be considered as bilinear forms.

### Why does the video mention non-standard notation for the tensor product?

-The video mentions non-standard notation for the tensor product to simplify the representation, omitting the circle-times operator and writing the covectors next to each other.

### What is the tensor product in the context of linear maps?

-The tensor product is a process that combines vectors and covectors together, allowing for a new perspective on linear maps where transformation rules are derived naturally.

### How does the new perspective on linear maps benefit the understanding of transformation rules?

-The new perspective allows for the transformation rules of linear maps to be obtained almost for free by simply transforming the basis vectors and covectors individually.

### What is the significance of the Kronecker delta in the context of linear maps acting on a vector?

-The Kronecker delta is used in the component multiplication formula to automatically get the correct components for the output vector when a linear map acts on a vector.

### Why are covector-covector pairs chosen for bilinear forms instead of vector-vector pairs?

-Covectors take one vector input each, so a pair of covectors naturally takes two vector inputs, which is suitable for bilinear forms that require two vector inputs.

### What is the transformation rule for the components of a bilinear form when written as a linear combination of covector-covector pairs?

-The transformation rule for the components is obtained by transforming the basis covectors individually using the forward transform F.

### How does the video explain the multiplication formula for a bilinear form acting on two vector inputs?

-The video explains that by replacing the bilinear form and the vectors with their linear combination expansions in some basis and applying the linearity of covectors and index cancellation rules, the correct component multiplication formula is obtained.

### What is the correct array shape for bilinear forms when viewed as a row of rows?

-The correct array shape for bilinear forms is a row of rows, which makes sense when considering the matrix multiplication formula where both vectors can be written as columns.

### What benefits does viewing bilinear forms as linear combinations of covector-covector pairs provide?

-This perspective provides the transformation rules, the component multiplication formula, and the correct array shape for bilinear forms, all derived naturally from the linear combinations.

###### Outlines

##### π Introduction to Covector-Covectors Pairs and Bilinear Forms

This paragraph introduces the concept of covector-covector pairs and their role in bilinear forms. The video explains that these pairs are, in fact, bilinear forms and highlights a non-standard notation for the tensor product, where the circle-times operator is omitted. The paragraph also revisits the concept of linear maps as linear combinations of vector-covector pairs and the benefits of this perspective, such as simplifying the transformation of linear maps between bases, deriving matrix-vector multiplication formulas, and understanding the array shape of tensors. The video emphasizes the utility of viewing bilinear forms through the lens of covector-covector pairs, which aligns with their function of taking two vector inputs.

##### π Transformation Rules and Benefits of Covector-Covectors Pairs

The second paragraph delves into the transformation rules for bilinear forms when expressed as linear combinations of covector-covector pairs. It explains the process of transforming basis covectors using the forward transform F, which is essential for understanding how bilinear forms change under different bases. The paragraph also discusses the component multiplication formula for bilinear forms acting on two vector inputs, demonstrating how to derive the correct components using linearity and the Kronecker delta index cancellation rule. Finally, it addresses the array shape of bilinear forms, contrasting the traditional matrix representation with the novel 'row of rows' perspective, which aligns with the natural columnar representation of vectors and simplifies the multiplication process.

###### Mindmap

###### Keywords

##### π‘Covector-covector pairs

##### π‘Bilinear forms

##### π‘Tensor product

##### π‘Linear map

##### π‘Basis vectors and covectors

##### π‘Transformation rules

##### π‘Matrix-vector multiplication

##### π‘Kronecker delta

##### π‘Array shape

##### π‘Distributive property

##### π‘Metric tensor

###### Highlights

Introduction of the concept of covector-covector pairs and their role in bilinear forms.

Use of non-standard notation for the tensor product by omitting the circle-times operator.

Linear maps can be represented as linear combinations of vector-covector pairs.

Benefits of the new perspective on linear maps include automatic transformation rules.

Transformation of basis vectors and covectors using backward and forward transforms.

Automatic derivation of matrix-vector component multiplication formula using linearity and Kronecker delta.

Tensor product's role in determining the correct array shape for tensors.

Distributing arrays into each other as a useful concept for understanding linear maps.

Perspective change for bilinear forms viewing them as linear combinations of covector-covector pairs.

Choice of covector-covector pairs for bilinear forms due to their ability to take two vector inputs.

Transformation rules for bilinear form components by individually transforming basis covectors.

Component multiplication formula for bilinear forms acting on two vector inputs.

Correct array shape for bilinear forms as a row of rows, contrasting with the traditional matrix view.

Understanding vectors as columns and the implications for bilinear form representation.

Matrix multiplication formula made clearer by viewing bilinear forms as rows of rows.

Summation of insights on writing bilinear forms as linear combinations of covector-covector pairs.

Automatic acquisition of transformation rules, component multiplication formula, and array shape.

###### Transcripts

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