Linear Algebra and Geometry 1

Systems of equations, matrices, determinants, vectors, and geometry of straight lines and planes in the 3-space

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Hania Uscka-Wehlou

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47.0 giờ video

223 bài giảng

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Học trên mọi thiết bị

Cập nhật: 12/06/2026

#Geometry #Linear Algebra #Mathematics #Vector Calculus

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How to solve problems in linear algebra and geometry (illustrated with 175 solved problems) and why these methods work.

Solve systems of linear equations with help of Gauss-Jordan or Gaussian elimination, the latter followed by back-substitution.

Interpret geometrically 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: determine whether a matrix is invertible; compute its inverse: both with (Jacobi) algorithm and by the explicit formula; matrix equations.

Determinants, their definition, properties, and different ways of computing them; determinant equations; Cramer's rule for n-by-n systems of equations.

Vectors, their coordinates and norm; geometrical vectors and abstract vectors, their addition and scaling: arithmetically and geometrically (in 2D and 3D).

Vector products (scaling, dot product, cross product, scalar triple product), their properties and applications; orthogonal projection 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 2D or planes in 3D.

Understand the connection between systems of linear equations and matrix multiplication.

Invertible Matrix Theorem and its applications; apply determinant test in various situations.

Yêu cầu

High-school maths, mainly arithmetic, some Trigonometry

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.

Đối tượng phù hợp

University and college engineering

Nội dung khóa học

21 phần • 223 bài giảng

1 Introduction

1 Coordinate systems and coordinates in the plane and in the 3-space.

2 Slope-intercept equations of straight lines in the plane

3 Normal equations of planes in the 3-space.

4 Vectors

5 Scalars

6 Vector addition and vector scaling

7 Linear combinations

8 Matrices

9 Linear transformations

10 Matrix—vector multiplication

11 Rules for computations with real numbers

12 Pythagorean Theorem and distance between points

13 Sine, cosine, and pythagorean identity

14 Cosine Rule

1 Different ways of looking at equations

2 Solution set

3 Linear and non-linear equations

4 Systems of linear equations

5 Solution sets of systems of linear equations

6 An example of a 2 × 2 system of linear equations, a graphical solution

7 Possible solution sets of 2 × 2 systems of linear equations

8 Possible solution sets of 3 × 2 systems of linear equations

9 Possible solution sets of 3 × 3 systems of linear equations

10 Possible solution sets of 2 × 3 systems of linear equations

11 Possible solution sets of m × n systems of linear equations

1 Our earlier problem revisited; an algebraical solution

2 Three elementary operations

3 What is Gauss—Jordan elimination and Gaussian elimination?

4 Gauss—Jordan elimination, a 2-by-2 system with unique solution

5 The same example solved with Gaussian elimination and back-substitution

6 The same example solved with matrix operations; coefficient matrix and augmented

7 How to write the augmented matrix for a given system of equations, Problem 1

8 How to write system of equations to a given augmented matrix, Problem 2

9 Gaussian elimination, Problem 3

10 Gaussian elimination, Problem 4

11 Gaussian elimination, Problem 5

12 Gaussian elimination, Problem 6.

13 What happens if the system is inconsistent?

14 Gaussian elimination, Problem 7.

15 Preparation to the general formulation of the algorithm; REF and RREF matrices.

16 How to read solutions from REF and RREF matrices?

17 General formulation of the algorithm in Gauss–Jordan elimination

18 Gauss–Jordan elimination, Problem 8

19 Gauss–Jordan elimination, Problem 9

20 Gaussian elimination, Problem 10

21 Gauss–Jordan elimination, Problem 11

22 Gauss–Jordan elimination, Problem 12

23 Gauss–Jordan elimination, Problem 13

1 Solving systems of linear equations in Linear Algebra and Geometry

2 Solving systems of linear equations (Calculus) Problem 1

3 Solving systems of linear equations (Calculus) Problem 2

4 (Optional from 22:30) Solving systems of linear equations (Calculus) Problem 3

5 Solving systems of linear equations (Calculus) Problem 4

6 Problem 5 (Chemistry)

7 Problem 6 (Electrical circuits)

1 Introduction to matrices

2 Different types of matrices

3 Matrix addition and subtraction, Problem 1

4 Matrix scaling, with geometrical interpretation

5 Matrix scaling, Problem 2

6 Matrix multiplication, with geometrical interpretation

7 Matrix multiplication, how to do

8 Matrix multiplication, Problem 3

9 Matrix multiplication and systems of equations, Problem 4

10 Transposed matrix, definition and some examples

11 Trace of a matrix, definition and an example

12 Various matrix operations, Problem 7

13 Various matrix operations, Problem 8

1 Properties of matrix operations, an introduction

2 Matrix addition has all the good properties

3 Matrix multiplication has a neutral element for square matrices

4 Matrix multiplication is associative

5 Matrix multiplication is not commutative

6 Sometimes commutativity happens, Problem 1

7 Two distributive laws

8 Matrix multiplication does not have the zero-product property

9 There is no cancellation law for matrix multiplication

10 Inverse matrices; not all non-zero square matrices have an inverse

11 Inverse matrix for 2-by-2 matrices; non-zero determinant

12 Solving matrix equations, Problem 2

13 Powers of matrices; powers of diagonal matrices

14 Computation rules for transposed matrices

15 Supplement to Video 83; Inverse of a product

16 Inverse of a transposed matrix

17 Various rules, Problem 3

1 Inverse matrices, introduction to the algorithm

2 Algorithm for inverse matrices, an example

3 Matrix inverse, Problem 1

4 Matrix inverse, Problem 2

5 Matrix equations, Problem 3

6 Matrix equations, Problem 4

7 Matrix equations, Problem 5

8 Matrix equations, Problem 6

9 Matrix inverse, Problem 7

10 Elementary operations and elementary matrices

11 Inverse elementary operations and their matrices

12 A really important theorem

13 Four equivalent statements

1 Formally about the number of solutions to systems of linear equations

2 Two more statements in our important theorem

3 Solution of a linear system using A inverse, Problem 1

4 Determining consistency by elimination, Problem 2

5 Matrix equations, Problem 3

1 Why the determinants are important

2 2-by-2 determinants; notation for n-by-n determinants

3 Geometrical interpretations of determinants

4 Geometrically about the determinant of a product

5 Definition of determinants

6 Conclusion 1: Determinant of matrices with interchanged columns

7 Conclusion 2: What happens when one column is a linear combination of others

8 Conclusion 3: About adding a multiple of a column to another column

9 Conclusion 4: Determinant of kA for any k ∈ R

10 Elementary column operations

11 How to compute 2-by-2 determinants from the definition

12 How to compute 3-by-3 determinants from the definition

13 Sarrus’ rule for 3-by-3 determinants

14 Determinant of transposed matrix; row operations

15 Evaluating determinants by cofactor expansion along rows or columns

16 Evaluating determinants by row or column reduction

17 Determinant of inverse

18 Properties of determinants, Problem 1

19 Properties of determinants, Problem 2

20 Properties of determinants, Problem 3

21 Determinant equations, Problem 4

22 Determinant equations, Problem 5

23 Determinant equations, Problem 6

24 Determinant equations, Problem 7

25 Invertible matrices, determinant test with a proof, Problem 8

26 Cramer’s rule, a proof, an example, and a geometrical interpretation

27 Cramer’s rule, Problem 9

28 Inverse matrix, an explicit formula

29 Invertible matrices, Problem 10

30 Problem 11, a large determinant

31 Problem 12, another large determinant

32 Problem 13: a trigonometric determinant

33 Problem 14: Vandermonde determinant

1 Vectors, a repetition

2 Computation rules for vector addition and scaling

3 Computations with vectors, Problem 1

4 Computations with vectors, Problem 2

5 Computations with vectors, Problem 3

6 Parallel vectors, Problem 4

7 Parallel vectors, Problem 5

8 Linear combinations, Problem 6

9 Linear combinations, Problem 7

10 Linear combinations, linear independence, Problem 8

11 Linear combinations, linear dependence, Problem 9

12 Area, Problem 10

13 Midpoint of a line segment, Problem 11

1 Norm of a vector, Problem 1

2 Properties of the norm

3 Distance between points, Problem 2

4 Unit vectors, how to normalize a vector

5 Unit vectors in given direction, Problem 3

1 Different products for vectors

2 Perpendicular straight lines and orthogonal vectors

3 Orthogonal projections

4 Definition of dot product for geometrical vectors

5 How to compute dot product, an example

6 Dot product for vectors in R^n; orthogonality and angles

7 Properties of dot product

8 Angles between vectors, Problem 1

9 Angles between vectors, Problem 2

10 How to find vector orthogonal to a given vector in the plane or in the 3-space

11 Orthogonal projections and decompositions

12 Orthogonal projections and decompositions, Problem 3

13 Orthogonal projections and decompositions, Problem 4

14 Tập hợp trực giao, Bài tập 5

15 Phép chiếu trực giao và phân tích, Bài tập 6

1 Tích có hướng, phần giới thiệu

2 Tích có hướng, định nghĩa

3 Ba tính chất của tích có hướng

4 Độ dài của tích có hướng hai vectơ

5 Thêm các tính chất của tích có hướng

6 Tích có hướng: Bài tập 1

7 Tích có hướng trong mặt phẳng

8 Tích hỗn hợp và thể tích

9 Tích hỗn hợp, Bài tập 2

10 Tính thẳng hàng trong mặt phẳng và tính đồng phẳng trong không gian 3 chiều

11 Phép kiểm tra định thức cho vectơ, Bài tập 3

1 Đường thẳng trong mặt phẳng, phần giới thiệu

2 Dạng hệ số góc - giao điểm và dạng giao điểm

3 Phương trình pháp tuyến

4 Phương trình tham số

5 Phương trình định thức

6 Đường thẳng trong mặt phẳng, Bài tập 5

1 Mặt phẳng trong không gian 3 chiều, phần giới thiệu

2 Phương trình pháp tuyến và phương trình giao điểm

3 Phương trình tham số

4 Từ tham số sang pháp tuyến

5 Từ pháp tuyến sang tham số; tìm một điểm và hai vectơ song song cho một mặt phẳng cho trước

6 Phương trình định thức

7 Mặt phẳng: Bài tập 6

1 Đường thẳng trong không gian 3 chiều, phần giới thiệu

2 Đường thẳng trong không gian 3 chiều, Bài tập 1

3 Đường thẳng trong không gian 3 chiều, Bài tập 2

4 Đường thẳng trong không gian 3 chiều, Bài tập 3

1 Liên thuộc 1: điểm và mặt phẳng

2 Liên thuộc 2: mặt phẳng và đường thẳng

3 Liên thuộc 3: điểm và đường thẳng

4 Các đối tượng song song và trực giao

5 Mặt phẳng song song, Bài tập 4

6 Mặt phẳng song song: Bài tập 5

7 Mặt phẳng trực giao: Bài tập 6

8 Mặt phẳng và đường thẳng, Bài tập 7

9 Mặt phẳng và đường thẳng, Bài tập 8

10 Mặt phẳng, Bài tập 9

11 Mặt phẳng, đường thẳng và hệ phương trình, Bài tập 10

1 Khoảng cách giữa các tập hợp, tổng quát

2 Khoảng cách từ điểm đến mặt phẳng

3 Khoảng cách từ điểm đến mặt phẳng, Bài tập 1

4 Khoảng cách từ điểm đến mặt phẳng, Bài tập 2

5 Khoảng cách từ điểm đến đường thẳng

6 Khoảng cách từ điểm đến đường thẳng, Bài tập 3

7 Khoảng cách từ điểm đến đường thẳng, Bài tập 4

8 Khoảng cách giữa các đường thẳng (chéo nhau)

9 Khoảng cách giữa các đường thẳng (chéo nhau), Bài tập 5

10 Khoảng cách giữa các đường thẳng (chéo nhau), Bài tập 6

1 Tổng kết Đại số Tuyến tính và Hình học 1

2 Đại số Tuyến tính và Hình học 2, một vài lời giới thiệu

3 Lời kết

1 Bài giảng thưởng

Giới thiệu khóa học

Linear Algebra and Geometry 1

Systems of equations, matrices, vectors, and geometry

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. Inverses; 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.

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

“001 Outline_Linear_Algebra_and_Geometry_1.pdf”

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