Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
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Lessons 

1) Introduction to matricesWhat a matrix is. How to add and subtract them.640 views

2) Matrix multiplication (part 1)Multiplying two 2x2 matrices.573 views

3) Matrix multiplication (part 2)More on multiplying matrices.473 views

4) Idea behind inverting a 2x2 matrix  Matrices  PrecalculusWhat the inverse of a matrix is. Examples of inverting a 2x2 matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...546 views

5) Classic video on inverting a 3x3 matrix part 1  Matrices  PrecalculusInverting a 3x3 matrix Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...585 views

6) Classic video on inverting a 3x3 matrix part 2  Matrices  PrecalculusUsing GaussJordan elimination to invert a 3x3 matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...622 views

7) Matrices to solve a system of equations  Matrices  PrecalculusUsing the inverse of a matrix to solve a system of equations.554 views

8) Matrices to solve a vector combination problem  Matrices  PrecalculusUsing matrices to figure out if some combination of 2 vectors can create a 3rd vector607 views

9) Singular matrices  Matrices  PrecalculusWhen and why you can't invert a matrix. Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/preca ...518 views

10) 3variable linear equations (part 1)Visual intuition of a 3variable linear equation.503 views

11) Solving 3 Equations with 3 UnknownsSolving 3 equations with 3 unknowns (old video from July 2008)558 views

12) Vector examples  Vectors and spaces  Linear AlgebraVisually understanding basic vector operations Practice this lesson yourself on KhanAcademy.org right now: https://www.khanacademy.org/math/linear ...572 views

13) Parametric representations of lines  Vectors and spaces  Linear AlgebraParametric Representations of Lines in R2 and R3467 views

14) Linear combinations and span  Vectors and spaces  Linear AlgebraUnderstanding linear combinations and spans of vectors530 views

15) Introduction to linear independence  Vectors and spaces  Linear AlgebraIntroduction to linear dependence and independence570 views

16) More on linear independence  Vectors and spaces  Linear AlgebraMore examples determining linear dependence or independence.593 views

17) Span and linear independence example  Vectors and spaces  Linear AlgebraDetermining whether 3 vectors are linearly independent and/or span R3530 views

18) Linear subspaces  Vectors and spaces  Linear AlgebraIntroduction to linear subspaces of Rn678 views

19) Basis of a subspace  Vectors and spaces  Linear AlgebraUnderstanding the definition of a basis of a subspace529 views

20) Vector dot product and vector length  Vectors and spaces  Linear AlgebraDefinitions of the vector dot product and vector length610 views

21) Proving vector dot product properties  Vectors and spaces  Linear AlgebraProving the "associative", "distributive" and "commutative" properties for vector dot products.574 views

22) Proof of the CauchySchwarz inequality  Vectors and spaces  Linear AlgebraProof of the CauchySchwarz Inequality804 views

23) Vector triangle inequality  Vectors and spaces  Linear AlgebraProving the triangle inequality for vectors in Rn537 views

24) Defining the angle between vectors  Vectors and spaces  Linear AlgebraIntroducing the idea of an angle between two vectors550 views

25) Defining a plane in R3 with a point and normal vector  Linear AlgebraDetermining the equation for a plane in R3 using a point on the plane and a normal vector640 views

26) Cross product introduction  Vectors and spaces  Linear AlgebraIntroduction to the cross product605 views

27) Proof: Relationship between cross product and sin of angle  Linear AlgebraProof: Relationship between the cross product and sin of angle between vectors607 views

28) Dot and cross product comparison/intuition  Vectors and spaces  Linear AlgebraDot and Cross Product Comparison/Intuition555 views

29) Matrices: Reduced row echelon form 1  Vectors and spaces  Linear AlgebraSolving a system of linear equations by putting an augmented matrix into reduced row echelon form590 views

30) Matrices: Reduced row echelon form 2  Vectors and spaces  Linear AlgebraAnother example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form740 views

31) Matrices: Reduced row echelon form 3  Vectors and spaces  Linear AlgebraAnd another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form591 views

32) Matrix vector products  Vectors and spaces  Linear AlgebraDefining and understanding what it means to take the product of a matrix and a vector552 views

33) Introduction to the null space of a matrix  Vectors and spaces  Linear AlgebraShowing that the Null Space of a Matrix is a valid Subspace666 views

34) Null space 2: Calculating the null space of a matrix  Linear AlgebraCalculating the null space of a matrix517 views

35) Null space 3: Relation to linear independence  Vectors and spaces  Linear AlgebraUnderstanding how the null space of a matrix relates to the linear independence of its column vectors513 views

36) Column space of a matrix  Vectors and spaces  Linear AlgebraIntroduction to the column space of a matrix539 views

37) Null space and column space basis  Vectors and spaces  Linear AlgebraFiguring out the null space and a basis of a column space for a matrix627 views

38) Visualizing a column space as a plane in R3  Vectors and spaces  Linear AlgebraDetermining the planar equation for a column space in R3538 views

39) Proof: Any subspace basis has same number of elements  Linear AlgebraProof: Any subspace basis has same number of elements559 views

40) Dimension of the null space or nullity  Vectors and spaces  Linear AlgebraDimension of the Null Space or Nullity546 views

41) Dimension of the column space or rank  Vectors and spaces  Linear AlgebraDimension of the Column Space or Rank616 views

42) Showing relation between basis cols and pivot cols  Linear AlgebraShowing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation618 views

43) Showing that the candidate basis does span C(A)  Vectors and spaces  Linear AlgebraShowing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).618 views

44) A more formal understanding of functions  Matrix transformations  Linear AlgebraA more formal understanding of functions503 views

45) Vector transformations  Matrix transformations  Linear AlgebraIntroduction to the notion of vector transformations573 views

46) Linear transformations  Matrix transformations  Linear AlgebraIntroduction to linear transformations589 views

47) Matrix vector products as linear transformations  Linear AlgebraMatrix Vector Products as Linear Transformations538 views

48) Linear transformations as matrix vector products  Linear AlgebraShowing how ANY linear transformation can be represented as a matrix vector product507 views

49) Image of a subset under a transformation  Matrix transformations  Linear AlgebraExploring what happens to a subset of the domain under a transformation573 views

50) im(T): Image of a transformation  Matrix transformations  Linear AlgebraShowing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation.636 views

51) Preimage of a set  Matrix transformations  Linear AlgebraDefinition of preimage of a set508 views

52) Preimage and kernel example  Matrix transformations  Linear AlgebraExample involving the preimage of a set under a transformation. Definition of kernel of a transformation.558 views

53) Sums and scalar multiples of linear transformations  Linear AlgebraSums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication.542 views

54) More on matrix addition and scalar multiplication  Linear AlgebraMore on Matrix Addition and Scalar Multiplication627 views

55) Linear transformation examples: Scaling and reflections  Linear AlgebraCreating scaling and reflection transformation matrices (which are diagonal)599 views

56) Linear transformation examples: Rotations in R2  Linear AlgebraLinear Transformation Examples: Rotations in R2465 views

57) Rotation in R3 around the xaxis  Matrix transformations  Linear AlgebraConstruction a rotation transformation in R3539 views

58) Unit vectors  Matrix transformations  Linear AlgebraWhat unit vectors are and how to construct them476 views

59) Introduction to projections  Matrix transformations  Linear AlgebraDetermining the projection of a vector on s line477 views

60) Expressing a projection on to a line as a matrix vector prod  Linear AlgebraExpressing a Projection on to a line as a Matrix Vector prod502 views

61) Compositions of linear transformations 1  Matrix transformations  Linear AlgebraIntroduction to compositions of Linear Transformations611 views

62) Compositions of linear transformations 2  Matrix transformations  Linear AlgebraProviding the motivation for definition of matrix products504 views

63) Matrix product examples  Matrix transformations  Linear AlgebraExample of taking the product of two matrices547 views

64) Matrix product associativity  Matrix transformations  Linear AlgebraShowing that matrix products are associative595 views

65) Distributive property of matrix products  Matrix transformations  Linear AlgebraShowing that matrix products are associative436 views

66) Introduction to the inverse of a function  Matrix transformations  Linear AlgebraIntroduction to the inverse of a function468 views

67) Proof: Invertibility implies a unique solution to f(x)=y  Linear AlgebraProof: Invertibility implies a unique solution to f(x)=y for all y in codomain of f.507 views

68) Surjective (onto) and injective (onetoone) functions  Linear AlgebraIntroduction to surjective and injective functions548 views

69) Relating invertibility to being onto and onetoone  Linear AlgebraRelating invertibility to being onto (surjective) and onetoone (injective)575 views

70) Determining whether a transformation is onto  Linear AlgebraDetermining whether a transformation is onto621 views

71) Exploring the solution set of Ax = b  Matrix transformations  Linear AlgebraExploring the solution set of Ax=b (non homogeneous equations)630 views

72) Matrix condition for onetoone trans  Matrix transformations  Linear AlgebraShowing that the rank of the of an mxn transformation matrix has to be an for the transformation to be onetoone (injective)485 views

73) Simplifying conditions for invertibility  Matrix transformations  Linear AlgebraShowing that a transformation is invertible if and only if rref(A) is equal to the identity matrix480 views

74) Showing that inverses are linear  Matrix transformations  Linear AlgebraShowing that inverse transformations are also linear495 views

75) Deriving a method for determining inverses  Matrix transformations  Linear AlgebraDetermining a method for constructing inverse transformation matrices531 views

76) Example of finding matrix inverse  Matrix transformations  Linear AlgebraExample of calculating the inverse of a matrix508 views

77) Formula for 2x2 inverse  Matrix transformations  Linear AlgebraFiguring out the formula for a 2x2 matrix. Defining the determinant.473 views

78) 3 x 3 determinant  Matrix transformations  Linear AlgebraDeterminants: Finding the determinant of a 3x3 matrix571 views

79) n x n determinant  Matrix transformations  Linear AlgebraDefining the determinant for nxn matrices. An example of a 4x4 determinant.576 views

80) Determinants along other rows/cols  Matrix transformations  Linear AlgebraFinding the determinant by going along other rows or columns472 views

81) Rule of Sarrus of determinants  Matrix transformations  Linear AlgebraA alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus)629 views

82) Determinant when row multiplied by scalar  Matrix transformations  Linear AlgebraThe determinant when a row is multiplied by a scalar710 views

83) (correction) scalar multiplication of row  Matrix transformations  Linear AlgebraCorrection of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinant534 views

84) Determinant when row is added  Matrix transformations  Linear AlgebraThe determinant when one matrix has a row that is the sum of the rows of other matrices (and every other term is identical in the 3 matrices)602 views

85) Duplicate row determinant  Matrix transformations  Linear AlgebraDeterminant of a matrix with duplicate rows598 views

86) Determinant after row operations  Matrix transformations  Linear AlgebraWhat happens to the determinant when we perform a row operation523 views

87) Upper triangular determinant  Matrix transformations  Linear AlgebraThe determinant of an upper triangular matrix501 views

88) Simpler 4x4 determinant  Matrix transformations  Linear AlgebraCalculating a 4x4 determinant by putting in in upper triangular form first.502 views

89) Determinant and area of a parallelogram  Matrix transformations  Linear AlgebraRealizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix606 views

90) Determinant as scaling factor  Matrix transformations  Linear AlgebraViewing the determinant of the transformation matrix as a scaling factor of regions524 views

91) Transpose of a matrix  Matrix transformations  Linear AlgebraTranspose of a matrix605 views

92) Determinant of transpose  Matrix transformations  Linear AlgebraProof by induction that transposing a matrix does not change its determinant516 views

93) Transpose of a matrix product  Matrix transformations  Linear AlgebraTaking the transpose of the product of two matrices516 views

94) Transposes of sums and inverses  Matrix transformations  Linear AlgebraTransposes of sums and inverses504 views

95) Transpose of a vector  Matrix transformations  Linear AlgebraTranspose of a column vector. Matrixmatrix products using vectors547 views

96) Rowspace and left nullspace  Matrix transformations  Linear AlgebraRowspace and Left Nullspace639 views

97) Visualizations of left nullspace and rowspace  Linear AlgebraRelationship between left nullspace, rowspace, column space and nullspace.498 views

98) Orthogonal complements  Alternate coordinate systems (bases)  Linear AlgebraOrthogonal Complements as subspaces.551 views

99) rank(a) = rank(transpose of a)  Matrix transformations  Linear AlgebraRank(A) = Rank(transpose of A)450 views

100) dim(v) + dim(orthogonal complement of v) = n  Linear AlgebraShowing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n497 views

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