This free online linear algebra course teaches introductory concepts in vectors and matrix algebra. The course consists of 56 tutorials which cover the material of a typical first year linear algebra course at the university level.

In order to gain a comprehensive understanding of the subject, you should start at the top and work your way down the list. That way you will start learning the most basic concepts first and build off of those as you progress through the course.

Vector Introduction

1. Real numbers and dimensions explained 
2. Draw a vector in standard position, or anywhere
3. Find the scalar multiple of a vector
4. Adding vectors
5. Subtracting vectors
6. Find the dot product of vectors
7. Find the length of a vector and give a unit vector in it's direction
8. Determine orthogonality and angles between vectors
9. Determine if the triangle formed by three points is a right-angled triangle
10. Describe all vectors [x, y] that are orthogonal to [a, b]
11. Find the Projection of v onto u
12. Proof that Proj(u)v = Proj(u)(Proj(u)v)
13. Proof that Proj(u)(v-Proj(u)v)=0
14. Write the equation of the line in general form, vector form, or parametric form
15. Find the distance from a point to a line
16. Determine the cross product of two vectors in R^3
17. Determine where two lines intersect

Systems of Linear Equations

18. Solve a system of linear equations using back substitution
19. Find the augmented matrix of a linear system of equations
20. Row echelon form of a matrix explained
21. Elementary row operations introduced and proved
22. Use elementary row operations to solve an augmented matrix
23. Solve an augmented matrix by finding its reduced row echelon form
24. Determine if a system of linear equations has one, none, or infinite solutions
25. How to solve Ax=b

Vector Combinations and Span

26. How to determine if one vector is a linear combination of a set of vectors 
27. How to determine if one vector in the span of a set of vectors 
28. Write a vector as a linear combination of a set of vectors 
29. How to determine if a set of vectors are linearly dependent or independent 
30. Write a dependence equation for a set of vectors 
31. Does a set of vectors span R^n
32. What is a subspace
33. Find a basis for the span of a set of vectors (either a subspace or a vector space)
34. Some thoughts on linear combinations and span
​

Matrices

35. The linear algebra matrix introduced
36. Matrix addition, scalar multiplication, and "subtraction"
37. Matrix multiplication
38. Matrix transpose
39. Find the determinant of a 2x2 matrix
40. Find the determinant of a 3x3 matrix using cofactor expansion
41. Find the determinant of a 3x3 matrix the fast way
42. Find the inverse of a 2x2 matrix
43. Find the inverse of a 3x3 matrix using the Gauss-Jordan method
​44. Find the nullspace of a matrix

Eigenvalues and Eigenvectors

45. Wtf are eigenvalues and eigenvectors 
46. Show that x is an eigenvector of A and find the corresponding eigenvalue
47. Show that λ is an eigenvalue of A and find a corresponding eigenvector
48. Find the characteristic polynomial of a matrix
49. Find the eigenvalues and eigenvectors of a 2x2 matrix
50. Find the eigenvalues and eigenvectors of a 3x3 matrix

Complex Numbers

51. A quick introduction to complex numbers
52. How to add and subtract complex numbers
53. How to multiply complex numbers
54. How to divide complex numbers
55. How to find the absolute value of a complex number
56. How to write a complex number in polar form

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