Tensors for Beginners 7: Linear Maps

eigenchris
24 Jan 201812:05
EducationalLearning
32 Likes 10 Comments

TLDRThis video script delves into the concept of linear maps, a type of tensor, as transformations taking vectors to new vectors. It introduces linear maps through three perspectives: coordinate representation as matrices, geometric interpretation preserving lines and origin, and an abstract algebraic definition emphasizing linearity properties. The script clarifies the matrix multiplication's connection to these properties, illustrating how linear maps work across different dimensions and the significance of basis vectors in transformations.

Takeaways
  • 📚 Linear maps are the third type of tensor discussed in the video, defined as transformations that take a vector as input and produce a new vector as output.
  • 📊 The coordinate representation of linear maps is through matrices, which are two-dimensional arrays of numbers.
  • 🔍 Matrices transform vectors using standard matrix multiplication rules, involving dot products with rows of the matrix.
  • 🌐 The geometrical interpretation of linear maps includes spatial transformations that preserve parallel lines, even spacing, and a stationary origin.
  • 🚫 Translations, despite keeping lines parallel and evenly spaced, are not considered linear maps because they move the origin.
  • 📝 The abstract definition of a linear map is a function that maps vectors to vectors, obeying the properties of linearity, such as additivity and homogeneity.
  • 🔑 The abstract properties of linear maps are demonstrated geometrically, showing that the transformation of the sum of vectors is equivalent to the sum of their transformations.
  • 🔄 The matrix multiplication rule is derived from the abstract linearity properties, explaining why the standard matrix multiplication formula works as it does.
  • 📏 Linear maps do not change the basis; they only transform vectors, and the output is measured using the same basis as the input.
  • 📈 The i-th column of a matrix tells you where the i-th basis vector is mapped to, providing an intuitive understanding of matrix columns.
  • 🔬 The video script aims to show the relationship between the coordinate, geometrical, and abstract definitions of linear maps, emphasizing their interconnectedness.
Q & A

  • What is a linear map in the context of tensors?

    -A linear map is a transformation that takes a vector as input and produces a new vector as output, maintaining the properties of linearity such as additivity and homogeneity.

  • How are linear maps represented in coordinate form?

    -In coordinate form, linear maps are represented by matrices, which are two-dimensional arrays of numbers that transform vectors according to matrix multiplication rules.

  • What is the geometrical interpretation of a matrix?

    -Geometrically, a matrix can be interpreted as a linear map that transforms vectors without changing the basis, where the i-th column of the matrix represents the image of the i-th basis vector under the transformation.

  • How do linear maps relate to the concept of basis vectors?

    -Linear maps transform vectors but do not alter the basis vectors themselves. They determine the new positions of the basis vectors' copies in the output space, which defines the transformation.

  • What properties must a spatial transform have to be considered a linear map?

    -A spatial transform must keep lines parallel, maintain even spacing between lines, and keep the origin stationary to be considered a linear map.

  • Why are translations not considered linear maps even though they keep gridlines parallel and evenly spaced?

    -Translations are not considered linear maps because they move the origin, which is not allowed in linear maps that must keep the origin stationary.

  • What is the abstract definition of a linear map?

    -Abstractly, a linear map is a function that maps vectors from one vector space to another while obeying the properties of additivity and homogeneity, often referred to as 'linearity'.

  • How do the abstract properties of linear maps relate to their geometrical interpretation?

    -The abstract properties of linear maps, such as the ability to add inputs or outputs and scale inputs or outputs to achieve the same result, directly correspond to the geometrical behavior of linear maps in preserving parallelism and even spacing of lines.

  • Can you explain how matrix multiplication rules are derived from the abstract properties of linear maps?

    -Matrix multiplication rules are derived from the abstract properties of linear maps by expressing the transformation of basis vectors in terms of the basis of the output space, leading to the coefficients that define matrix multiplication.

  • What happens to the transformation rules of linear maps when we change the basis?

    -When the basis is changed, the transformation rules of linear maps are affected as the matrix representation of the linear map will change to reflect the new basis, although the underlying linear map itself remains the same.

  • Why is the abstract definition of a linear map considered the 'best' or 'truest' definition?

    -The abstract definition is considered the best because it captures the essence of linear maps in a way that is independent of any specific coordinate system or geometric representation, making it universally applicable across different contexts.

Outlines
00:00
📚 Introduction to Linear Maps and Matrices

This paragraph introduces the concept of linear maps as the third example of a tensor. Linear maps are defined as transformations that take a vector as input and produce a new vector as output. The speaker outlines three definitions of linear maps: a coordinate definition as arrays of numbers, a geometrical definition in terms of visual representation, and an abstract algebraic definition. The coordinate representation is explained through matrices, which are two-dimensional arrays of numbers, and the process of matrix multiplication is described. The geometrical interpretation is given through the effect of linear maps on basis vectors, emphasizing that the basis itself does not change during transformation. The paragraph concludes with a visual example of how matrices map basis vectors to new vectors.

05:04
📐 Geometric Interpretation of Linear Maps

The second paragraph delves into the geometric definition of linear maps, which are spatial transformations that preserve the parallelism of lines, maintain even spacing, and keep the origin fixed. The speaker provides visual examples of different types of linear maps, including stretching, rotation, and skew transformations, all of which adhere to the properties of linear maps. It is clarified that translations, despite sharing some characteristics, are not considered linear maps because they move the origin. The paragraph also connects the abstract definition of linear maps to their geometric interpretation, showing how the algebraic properties of linearity manifest in geometric transformations.

10:04
🔍 Abstract Definition and Relation to Matrix Multiplication

In this paragraph, the abstract definition of linear maps is presented as functions that map vectors to vectors while obeying the properties of linearity, such as the ability to add inputs or outputs and scale inputs or outputs without changing the result. The speaker then demonstrates how the abstract properties of linear maps relate to the geometric definitions and how they lead to the rules of matrix multiplication. By expanding a vector into its components and applying the linearity properties, the derivation of the matrix multiplication formula is shown, illustrating that the abstract algebraic properties are the foundation of the coordinate representation of linear maps.

Mindmap
Keywords
💡Covectors
Covectors are mathematical objects that generalize the concept of linear functionals, which take a vector as input and produce a scalar (a single number) as output. In the video, covectors are mentioned as a precursor to discussing linear maps, emphasizing the transition from scalar outputs to vector outputs in the context of tensors.
💡Linear Maps
Linear maps, also known as linear transformations, are the main subject of the video. These are functions that take a vector as input and produce a new vector as output, following the principles of linearity, which include the ability to add inputs and scale inputs without changing the output's form. The script uses linear maps to explain how vectors are transformed while preserving the structure of the vector space.
💡Coordinate Definition
A coordinate definition is an approach to understanding mathematical objects by representing them with arrays of numbers. In the context of the video, linear maps are introduced with a coordinate representation, which are matrices, a two-dimensional array of numbers that can be used to transform vectors according to specific rules.
💡Matrices
Matrices are the coordinate representation of linear maps. They are used to perform transformations on vectors through matrix multiplication. The video script explains how a matrix, acting on a column vector, can produce an output vector, illustrating the process with a 2-by-2 matrix example.
💡Basis Vectors
Basis vectors are fundamental vectors in a vector space that can be combined through linear operations to create any vector in that space. The script mentions basis vectors in the context of linear maps, explaining that while vectors are transformed, the basis vectors themselves remain unchanged.
💡Geometric Definition
The geometric definition of linear maps, as discussed in the video, refers to the visual and spatial interpretation of these transformations. It involves properties such as keeping lines parallel, maintaining even spacing between lines, and keeping the origin stationary, which are all characteristics of linear transformations.
💡Abstract Definition
An abstract definition provides a high-level, conceptual understanding of a mathematical object without reference to specific coordinates or geometric interpretations. In the video, the abstract definition of a linear map is presented as a function mapping vectors to vectors while obeying the principles of linearity, which includes the ability to add inputs and scale inputs and outputs.
💡Linearity
Linearity is a property of functions that states they can be modeled by linear equations, meaning they preserve scalar multiplication and vector addition. In the script, linearity is a key concept that defines both covectors and linear maps, with the distinction that linear maps output vectors instead of scalars.
💡Matrix Multiplication
Matrix multiplication is the process of transforming a vector using a matrix. The video explains how this process is derived from the abstract properties of linear maps, showing that the standard matrix multiplication formula is a direct consequence of the linearity properties.
💡Transformation Rules
Transformation rules refer to the specific ways in which linear maps alter vectors. The video script hints at discussing how these rules change when the basis of the vector space is altered, which is a fundamental concept in understanding how linear maps behave under different representations of vector spaces.
Highlights

Introduction to linear maps as the third example of a tensor, defining them as transformations taking vectors to new vectors.

Coordinate definition of linear maps as arrays of numbers, specifically matrices.

Geometric definition of linear maps in terms of spatial transformations.

Abstract definition of linear maps as purely algebraic functions mapping vectors to vectors.

Matrix multiplication rules applied to transform vectors.

Interpretation of matrix columns as outputs for basis vector inputs.

Explanation of basis vectors' immutability under linear map transformations.

Visual representation of linear maps with matrices affecting vectors but not the basis.

Geometric properties of linear maps: preserving parallel lines, even spacing, and stationary origin.

Differentiation between linear maps and translations in terms of origin movement.

Abstract properties of linear maps involving input and output addition and scaling.

Demonstration of algebraic properties of linear maps through geometric illustrations.

Derivation of matrix multiplication from abstract linearity properties.

Explanation of how matrix coefficients relate to basis vectors and linear combinations.

Generalization of the matrix multiplication formula for any number of dimensions.

Connection between abstract linearity and the mechanics of matrix multiplication.

Upcoming discussion on transformation rules of linear maps under basis changes.

Transcripts

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Related Tags
Linear MapsTensorsMatrix MultiplicationVector TransformationAbstract AlgebraGeometric InterpretationBasis VectorsLinearity PropertiesSpatial TransformsCoordinate Systems