# Linear Algebra and Geometry 2

## What you'll learn

- How to solve problems in linear algebra and geometry (illustrated with 153 solved problems) and why these methods work.
- Important concepts concerning vector spaces, such as basis, dimension, coordinates, and subspaces.
- Linear combinations, linear dependence and independence in various vector spaces, and how to interpret them geometrically in R^2 and R^3.
- How to recalculate coordinates from one basis to another, both with help of transition matrices and by solving systems of equations.
- Row space, columns space and nullspace for matrices, and about usage of these concepts for solving various types of problems.
- Linear transformations: different ways of looking at them (as matrix transformations, as transformations preserving linear combinations).
- How to compose linear transformations and how to compute their standard matrices in different bases; compute the kernel and the image for transformations.
- Understand the connection between matrices and linear transformations, and see various concepts in accordance with this connection.
- Work with various geometrical transformations in R^2 and R^3, be able to compute their matrices and explain how these transformations work.
- Understand the concept of isometry and be able to give some examples, and formulate their connection with orthogonal matrices.
- Transform any given basis for a subspace of R^n into an orthonormal basis of the same subspace with help of Gram-Schmidt Process.
- Compute eigenvalues, eigenvectors, and eigenspaces for a given matrix, and give geometrical interpretations of these concepts.
- Determine whether a given matrix is diagonalizable or not, and perform its diagonalization if it is.
- Understand the relationship between diagonalizability and dimensions of eigenspaces for a matrix.
- Use diagonalization for problem solving involving computing the powers of square matrices, and motivate why this method works.
- Be able to formulate and use The Invertible Matrix Theorem and recognise the situations which are suitable for the determinant test (and which are not).
- Use Wronskian to determine whether a set of smooth functions is linearly independent or not; be able to compute Vandermonde determinant.
- Work with various vector spaces, for example with R^n, the space of all n-by-m matrices, the space of polynomials, the space of smooth functions.

## Requirements

- Linear Algebra and Geometry 1 (systems of equations, matrices and determinants, vectors and their products, analytic geometry of lines and planes)
- High-school and college mathematics (mainly arithmetic, some Trigonometry, polynomials)
- Some basic Calculus (used in some examples that can be omitted without having any negative impact on your studies of Linear Algebra, and revisited after you have studied Calculus: some simple derivatives are needed in Videos 38, 39, 40, 132, 134, 136; continuous functions are mentioned in Videos 16, 35--40)
- Basic knowledge of complex numbers (used in an example in Video 8)
- You are always welcome with your questions. If something in the lectures is unclear, please, ask. It is best to use QA, so that all the other students can see my additional explanations about the unclear topics. Remember: you are never alone with your doubts, and it is to everybody's advantage if you ask your questions on the forum.

## Description

Linear Algebra and Geometry 2

*Much more about matrices; abstract vector spaces and their bases*

Chapter 1: Abstract vector spaces and related stuff

S1. Introduction to the course

S2.** **Real vector spaces and their subspaces

You will learn: the definition of vector spaces and the way of reasoning around the axioms; determine whether a subset of a vector space is a subspace or not.

S3.** **Linear combinations and linear independence

You will learn: the concept of linear combination and span, linearly dependent and independent sets; apply Gaussian elimination for determining whether a set is linearly independent; geometrical interpretation of linear dependence and linear independence.

S4. Coordinates, basis, and dimension

You will learn: about the concept of basis for a vector space, the coordinates w.r.t.\ a given basis, and the dimension of a vector space; you will learn how to apply the determinant test for determining whether a set of *n* vectors is a basis of R^n.

S5. Change of basis

You will learn: how to recalculate coordinates between bases by solving systems of linear equations, by using transition matrices, and by using Gaussian elimination; the geometry behind different coordinate systems.

S6. Row space, column space, and nullspace of a matrix

You will learn: concepts of row and column space, and the nullspace for a matrix; find bases for span of several vectors in R^n with different conditions for the basis.

S7. Rank, nullity, and four fundamental matrix spaces

You will learn: determine the rank and the nullity for a matrix; find orthogonal complement to a given subspace; four fundamental matrix spaces and the relationship between them.

Chapter 2: Linear transformations

S8. Matrix transformations from R^n to R^m

You will learn: about matrix transformations: understand the way of identifying linear transformations with matrices (produce the standard matrix for a given transformation, and produce the transformation for a given matrix); concepts: kernel, image and inverse operators; understand the link between them and nullspace, column space and inverse matrix.

S9. Geometry of matrix transformations on R^2 and R^3

You will learn: about transformations such as rotations, symmetries, projections and their matrices; you will learn how to illustrate the actions of linear transformations in the plane.

S10. Properties of matrix transformations

You will learn: what happens with subspaces and affine spaces (points, lines and planes) under linear transformations; what happens with the area and volume; composition of linear transformations as matrix multiplication.

S11. General linear transformations in different bases

You will learn: solving problems involving linear transformations between two vector spaces; work with linear transformations in different bases.

Chapter 3: Orthogonality

S12. Gram-Schmidt Process

You will learn: about orthonormal bases and their superiority above the other bases; about orthogonal projections on subspaces to R^n; produce orthonormal bases for given subspaces of R^n with help of Gram-Schmidt process.

S13. Orthogonal matrices

You will learn: definition and properties of orthonormal matrices; their geometrical interpretation.

Chapter 4: Intro to eigendecomposition of matrices

S14. Eigenvalues and eigenvectors

You will learn: compute eigenvalues and eigenvectors for square matrices with real entries; geometric interpretation of eigenvectors and eigenspaces.

S15. Diagonalization

You will learn: to determine whether a given matrix is diagonalizable or not; diagonalize matrices and apply the diagonalization for problem solving (the powers of matrices).

S16. Wrap-up Linear Algebra and Geometry 2

You will learn: about the content of the third course.

*Make sure that you check with your professor what parts of the course you will need for your final exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.*

*A detailed description of the content of the course, with all the 214 videos and their titles, and with the texts of all the 153 problems solved during this course, is presented in the resource file *

*"001 List_of_all_Videos_and_Problems_Linear_Algebra_and_Geometry_2.pdf” *

*under video 1 ("Introduction to the course"). This content is also presented in video 1.*

## Who this course is for:

- University and college engineering

## Instructors

I am a multilingual mathematician with a passion for mathematics education. I always try to find the simplest possible explanations for mathematical concepts and theories, with illustrations whenever possible, and with geometrical motivations.

I worked as a senior lecturer in mathematics at Uppsala University (from August 2017 to August 2019) and at Mälardalen University (from August 2019 to May 2021) in Sweden, but I terminated my permanent employment to be able to create courses for Udemy full-time.

I am originally from Poland where I studied theoretical mathematics and got pedagogical qualifications at the Copernicus University in Torun (1992-1997). Before that, I enjoyed a very rigorous mathematical education in a mathematical class in high school "Liceum IV" in Torun, which gave me a very solid foundation for everything else I have learned and taught later.

My PhD thesis (2009) was at Uppsala University in Sweden, with the title: "Digital Lines, Sturmian Words, and Continued Fractions".

In 2018 I received four pedagogical prizes from students at the Faculty of Science and Technology of Uppsala University: on May 13th from the students at the Master Program in Engineering Physics; on May 25th from the students at the Master Program in Electrical Engineering; on December 20th from the students at the Master Program in Chemical Engineering; on January 10th 2019 from UTN (Uppsala Union of Engineering and Science Students at Uppsala University).

I speak Polish, Swedish, English, Dutch, and some Russian; learning Ukrainian.

I have a background in medicine and software development. I've done enough mathematics to at least follow along in Hania's courses and I'm learning a lot as I edit the material. I have also written a book about medical software design as it pertains to the medical record ("Rethinking the electronic healthcare record"). For Hania's math courses, it's my job to set up the environment and produce the final output that goes into these courses.