Linear Algebra: Orthogonality and Diagonalization

Linear Algebra: Orthogonality and Diagonalization

本课程是 Linear Algebra from Elementary to Advanced 专项课程的一部分

位教师：Joseph W. Cutrone, PhD

顶尖授课教师

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4个模块

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作业

11 项作业

授课语言：英语（English）

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本课程是 Linear Algebra from Elementary to Advanced 专项课程专项课程的一部分

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该课程共有4个模块

This is the third and final course in the Linear Algebra Specialization that focuses on the theory and computations that arise from working with orthogonal vectors. This includes the study of orthogonal transformation, orthogonal bases, and orthogonal transformations. The course culminates in the theory of symmetric matrices, linking the algebraic properties with their corresponding geometric equivalences. These matrices arise more often in applications than any other class of matrices.

The theory, skills and techniques learned in this course have applications to AI and machine learning. In these popular fields, often the driving engine behind the systems that are interpreting, training, and using external data is exactly the matrix analysis arising from the content in this course. Successful completion of this specialization will prepare students to take advanced courses in data science, AI, and mathematics.

In this module, we define a new operation on vectors called the dot product. This operation is a function that returns a scalar related to the angle between the vectors, distance between vectors, and length of vectors. After working through the theory and examples, we hone in on both unit (length one) and orthogonal (perpendicular) vectors. These special vectors will be pivotal in our course as we start to define linear transformations and special matrices that use only these vectors.

涵盖的内容

2个视频2篇阅读材料3个作业

In this module we will study the special type of transformation called the orthogonal projection. We have already seen the formula for the orthogonal projection onto a line so now we generalize the formula to the case of projection onto any subspace W. The formula will require basis vectors that are both orthogonal and normalize and we show, using the Gram-Schmidt Process, how to meet these requirements given any non-empty basis.

涵盖的内容

3个视频3篇阅读材料4个作业

3个视频总计51分钟

Orthogonal Projections 18分钟
Gram-Schmidt Process 14分钟
Least-Squares Problems 19分钟

3篇阅读材料总计30分钟

Orthogonal Projections 10分钟
Finding Orthogonal Bases 10分钟
Least-Squares Solutions 10分钟

4个作业总计120分钟

Orthogonal Projections Practice 30分钟
Orthogonal Bases Practice 30分钟
Least-Squares Solutions Practice 30分钟
Orthogonal Projections and Least Squares 30分钟

In this module we look to diagonalize symmetric matrices. The symmetry displayed in the matrix A turns out to force a beautiful relationship between the eigenspaces. The corresponding eigenspaces turn out to be mutually orthogonal. After normalizing, these orthogonal eigenvectors give a very special basis of R^n with extremely useful applications to data science, machine learning, and image processing. We introduce the notion of quadratic forms, special functions of degree two on vectors , which use symmetric matrices in their definition. Quadratic forms are then completely classified based on the properties of their eigenvalues.