This free online linear algebra course teaches introductory concepts in vectors and matrix algebra. The course consists of 68 tutorials which cover the material of a typical first year linear algebra course at the university level. Tutorials 125 are complete and tutorials 2668 are currently under development.
The lessons build on each other so we recommend that you start at the top of the list and watch them all in order. 
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 rightangled 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)(vProj(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. Is one vector a linear combination of a set of vectors?
27. Is one vector in the span of a set of vectors? 28. Write a vector as a linear combination of a set of vectors 29. Is a set of vectors linearly dependent or independent? 30. Write a dependence equation for a set of vectors 31. Does a set of vectors span R^n? 32. Find a basis for the span of a set of vectors 33. How you can travel through the fourth dimension without ever moving! Matrices34. The linear algebra matrix explained
35. Matrix addition, scalar multiplication, and "subtraction" 36. Matrix multiplication 37. Matrix Properties Explained 38. Find A^n 39. Write A as a linear combination of B, C 40. Find the inverse of a 2x2 matrix 41. Use the GaussJordan method to find the inverse of a 3x3 matrix 42. Determine if a vector is in the column space or row space of a matrix 43. Give a basis for row(A), col(A), and null(A) of matrix A 44. Find the determinant using cofactor expansion 45. Find the determinant quicker with shorthand method 46. Find the determinant using elementary row operations 47. Proof that B=A^1 if AB=I 48. Elementary matrices explained 49. Find the inverse of a matrix using elementary matrices Eigenvalues and Eigenvectors 50. Show that v is an eigenvector of A and find the corresponding eigenvalue
51. Show that λ is an eigenvalue of A and find a corresponding eigenvector 52. Find the characteristic polynomial of A using cofactor expansion 53. Find the characteristic polynomial of A quicker using shorthand notation 54. Find the eigenvalues and a basis for each eigenspace of A 55. Given eigen values, eigen vectors, and a vector x, solve for (A^n)(x) Orthogonality 56. Determine if the given vectors form an orthogonal set
57. Determine if the given vectors form an orthogonal basis for R^n 58. Determine if the given matrix is orthogonal 59. Determine if the given set of vectors is orthonormal 60. How to obtain an orthonormal basis for R^n 61. Orthogonally diagonalize a matrix 62. Given eigenvalues and orthogonal eigenvectors, find the symmetric matrix A 63. Solve a system of equations using Cramers rule Complex Numbers64. How to add and subtract complex numbers
65. How to multiply complex numbers 66. How to divide complex numbers 67. How to find the absolute value of a complex number 68. How to write a complex number in polar form 
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