# Linear Algebra and Geometry 1

## What you'll learn

- How to solve problems in linear algebra and geometry (illustrated with 175 solved problems) and why these methods work.
- Solve systems of linear equation with help of Gauss-Jordan or Gaussian elimination, the latter followed by back-substitution.
- Interpret geometrically the solution sets of systems of linear equations by analysing their RREF matrix (row equivalent with the augmented matrix of the system).
- Matrix operations (addition, scaling, multiplication), how they are defined, how they are applied, and what computational rules hold for them.
- Matrix inverse, how to determine whether a (square) matrix is invertible, and how to compute its inverse, both with (Jacobi) algorithm and by the explicit formula; solve matrix equations.
- Determinants, their definition, properties, and different ways of computing them; solve determinant equations; Cramer's rule for n-by-n systems of equations with invertible coefficient matrix.
- Vectors, their coordinates and norm; geometrical vectors (in the plane and in the space) and abstract vectors (in R^n), their addition and scaling, both arithmetically and geometrically (in R^2 and R^3).
- Various vector products (scaling, dot product, cross product, scalar triple product), their properties and domains of application; orthogonal projections and vector decomposition.
- Analytical geometry in the 3-space: different ways of describing lines and planes, with applications in problem solving.
- Compute distances between points, planes and lines in the 3-space, both by using orthogonal projections and by geometrical reasoning.
- Determine whether lines and planes are parallel, and compute the angles between them (using dot product and directional or normal vectors) if they intersect.
- How to geometrically interpret n-by-2 and n-by-3 systems of equations and their solution sets as intersection sets between lines in the plane and planes in the space.
- Understand the connection between systems of linear equations and matrix multiplication.
- Invertible Matrix Theorem and its applications; apply determinant test in various situations.

## Requirements

- High-school maths, mainly arithmetics, some trigonometry

## Description

Chapter 1 **Systems of linear equations**

S1. **Introduction to the course**

S2. **Some basic concepts**

You will learn: some basic concepts that will be used in this course. Most of them are known from high-school courses in mathematics, some of them are new; the latter will appear later in the course and will be treated more in depth then.

S3. **Systems of linear equations; building up your geometrical intuition**

You will learn: some basic concepts about linear equations and systems of linear equations; geometry behind systems of linear equations.

S4. **Solving systems of linear equations; Gaussian elimination**

You will learn: solve systems of linear equations using Gaussian elimination (and back-substitution) and Gauss--Jordan elimination in cases of systems with unique solutions, inconsistent systems, and systems with infinitely many solutions (parameter solutions).

S5. **Some applications in mathematics and natural sciences**

You will learn: how systems of linear equations are used in other branches of mathematics and in natural sciences.

Chapter 2 **Matrices and determinants**

S6. **Matrices and matrix operations**

You will learn: the definition of matrices and their arithmetic operations (matrix addition, matrix subtraction, scalar multiplication, matrix multiplication). Different kinds of matrices (square matrices, triangular matrices, diagonal matrices, zero matrices, identity matrix).

S7. I**nverses; Algebraic properties of matrices**

You will learn: use matrix algebra; the definition of the inverse of a matrix.

S8. **Elementary matrices and a method for finding A inverse**

You will learn: how to compute the inverse of a matrix with Gauss--Jordan elimination (Jacobi’s method).

S9. **Linear systems and matrices**

You will learn: about the link between systems of linear equations and matrix multiplication.

S10. **Determinants**

You will learn: the definition of the determinant; apply the laws of determinant arithmetics, particularly the multiplicative property and the expansion along a row or a column; solving equations involving determinants; the explicite formula for solving of n-by-n systems of linear equations (Cramer's rule), the explicite formula for inverse to a non-singular matrix.

Chapter 3 **Vectors and their products**

S11. **Vectors in 2-space, 3-space, and n-space**

You will learn: apply and graphically illustrate the arithmetic operations for vectors in the plane; apply the arithmetic operations for vectors in R^n.

S12. **Distance and norm in R^n**

You will learn: compute the distance between points in R^n and norms of vectors in R^n, normalize vectors.

S13. **Dot product, orthogonality, and orthogonal projections**

You will learn: definition of dot product and the way you can use it for computing angles between geometrical vectors.

S14. **Cross product, parallelograms and parallelepipeds**

You will learn: definition of cross product and interpretation of 3-by-3 determinants as the volume of a parallelepiped in the 3-space.

Chapter 4 **Analytical geometry of lines and planes**

S15. **Lines in R^2**

You will learn: several ways of describing lines in the plane (slope-intercept equation, intercept form, point-vector equation, parametric equation) and how to compute other kinds of equations given one of the equations named above.

S16. **Planes in R^3**

You will learn: several ways of describing planes in the 3-spaces (normal equation, intercept form, parametric equation) and how to compute other kinds of equations given one of the equations named above.

S17. **Lines in R^3**

You will learn: several ways of describing lines in the 3-space (point-vector equation, parametric equation, standard equation) and how to compute other kinds of equations given one of the equations named above.

S18. **Geometry of linear systems; incidence between lines and planes**

You will learn: determine the equations for a line and a plane and how to use these for computing intersections by solving systems of equations.

S19. **Distance between points, lines, and planes**

You will learn: determine the equations for a line and a plane and how to use these for computing distances.

S20. **Some words about the next course**

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

*Also 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 **222** videos and their titles, and with the texts of all the **175** problems solved during this course, is presented in the resource file “Outline_Linear_Algebra_and_Geometry_1.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

## Course content

- Preview14:38

## Instructors

Hania is an award-winning university teacher in mathematics, with teaching qualifications and a PhD in mathematics. She 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 she terminated her permanent employment to be able to develop courses for Udemy full-time.

Hania is originally from Poland where she studied theoretical mathematics at the Copernicus University in Torun (1992-1997).

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

In 2018 Hania 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).

Hania speaks Polish, Swedish, English, Dutch, and some Russian.

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.