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.594 views

2) Matrix multiplication (part 1)Multiplying two 2x2 matrices.532 views

3) Matrix multiplication (part 2)More on multiplying matrices.432 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 ...503 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 ...537 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 ...572 views

7) Matrices to solve a system of equations  Matrices  PrecalculusUsing the inverse of a matrix to solve a system of equations.519 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 vector572 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 ...479 views

10) 3variable linear equations (part 1)Visual intuition of a 3variable linear equation.464 views

11) Solving 3 Equations with 3 UnknownsSolving 3 equations with 3 unknowns (old video from July 2008)520 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 ...525 views

13) Parametric representations of lines  Vectors and spaces  Linear AlgebraParametric Representations of Lines in R2 and R3425 views

14) Linear combinations and span  Vectors and spaces  Linear AlgebraUnderstanding linear combinations and spans of vectors476 views

15) Introduction to linear independence  Vectors and spaces  Linear AlgebraIntroduction to linear dependence and independence520 views

16) More on linear independence  Vectors and spaces  Linear AlgebraMore examples determining linear dependence or independence.537 views

17) Span and linear independence example  Vectors and spaces  Linear AlgebraDetermining whether 3 vectors are linearly independent and/or span R3466 views

18) Linear subspaces  Vectors and spaces  Linear AlgebraIntroduction to linear subspaces of Rn632 views

19) Basis of a subspace  Vectors and spaces  Linear AlgebraUnderstanding the definition of a basis of a subspace481 views

20) Vector dot product and vector length  Vectors and spaces  Linear AlgebraDefinitions of the vector dot product and vector length574 views

21) Proving vector dot product properties  Vectors and spaces  Linear AlgebraProving the "associative", "distributive" and "commutative" properties for vector dot products.528 views

22) Proof of the CauchySchwarz inequality  Vectors and spaces  Linear AlgebraProof of the CauchySchwarz Inequality744 views

23) Vector triangle inequality  Vectors and spaces  Linear AlgebraProving the triangle inequality for vectors in Rn492 views

24) Defining the angle between vectors  Vectors and spaces  Linear AlgebraIntroducing the idea of an angle between two vectors501 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 vector597 views

26) Cross product introduction  Vectors and spaces  Linear AlgebraIntroduction to the cross product563 views

27) Proof: Relationship between cross product and sin of angle  Linear AlgebraProof: Relationship between the cross product and sin of angle between vectors567 views

28) Dot and cross product comparison/intuition  Vectors and spaces  Linear AlgebraDot and Cross Product Comparison/Intuition506 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 form549 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 form683 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 form546 views

32) Matrix vector products  Vectors and spaces  Linear AlgebraDefining and understanding what it means to take the product of a matrix and a vector515 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 Subspace624 views

34) Null space 2: Calculating the null space of a matrix  Linear AlgebraCalculating the null space of a matrix481 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 vectors468 views

36) Column space of a matrix  Vectors and spaces  Linear AlgebraIntroduction to the column space of a matrix497 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 matrix588 views

38) Visualizing a column space as a plane in R3  Vectors and spaces  Linear AlgebraDetermining the planar equation for a column space in R3495 views

39) Proof: Any subspace basis has same number of elements  Linear AlgebraProof: Any subspace basis has same number of elements506 views

40) Dimension of the null space or nullity  Vectors and spaces  Linear AlgebraDimension of the Null Space or Nullity505 views

41) Dimension of the column space or rank  Vectors and spaces  Linear AlgebraDimension of the Column Space or Rank575 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 equation583 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).579 views

44) A more formal understanding of functions  Matrix transformations  Linear AlgebraA more formal understanding of functions477 views

45) Vector transformations  Matrix transformations  Linear AlgebraIntroduction to the notion of vector transformations547 views

46) Linear transformations  Matrix transformations  Linear AlgebraIntroduction to linear transformations562 views

47) Matrix vector products as linear transformations  Linear AlgebraMatrix Vector Products as Linear Transformations507 views

48) Linear transformations as matrix vector products  Linear AlgebraShowing how ANY linear transformation can be represented as a matrix vector product481 views

49) Image of a subset under a transformation  Matrix transformations  Linear AlgebraExploring what happens to a subset of the domain under a transformation536 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.599 views

51) Preimage of a set  Matrix transformations  Linear AlgebraDefinition of preimage of a set483 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.515 views

53) Sums and scalar multiples of linear transformations  Linear AlgebraSums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication.498 views

54) More on matrix addition and scalar multiplication  Linear AlgebraMore on Matrix Addition and Scalar Multiplication601 views

55) Linear transformation examples: Scaling and reflections  Linear AlgebraCreating scaling and reflection transformation matrices (which are diagonal)556 views

56) Linear transformation examples: Rotations in R2  Linear AlgebraLinear Transformation Examples: Rotations in R2417 views

57) Rotation in R3 around the xaxis  Matrix transformations  Linear AlgebraConstruction a rotation transformation in R3507 views

58) Unit vectors  Matrix transformations  Linear AlgebraWhat unit vectors are and how to construct them441 views

59) Introduction to projections  Matrix transformations  Linear AlgebraDetermining the projection of a vector on s line446 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 prod467 views

61) Compositions of linear transformations 1  Matrix transformations  Linear AlgebraIntroduction to compositions of Linear Transformations574 views

62) Compositions of linear transformations 2  Matrix transformations  Linear AlgebraProviding the motivation for definition of matrix products466 views

63) Matrix product examples  Matrix transformations  Linear AlgebraExample of taking the product of two matrices521 views

64) Matrix product associativity  Matrix transformations  Linear AlgebraShowing that matrix products are associative552 views

65) Distributive property of matrix products  Matrix transformations  Linear AlgebraShowing that matrix products are associative405 views

66) Introduction to the inverse of a function  Matrix transformations  Linear AlgebraIntroduction to the inverse of a function437 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.472 views

68) Surjective (onto) and injective (onetoone) functions  Linear AlgebraIntroduction to surjective and injective functions500 views

69) Relating invertibility to being onto and onetoone  Linear AlgebraRelating invertibility to being onto (surjective) and onetoone (injective)547 views

70) Determining whether a transformation is onto  Linear AlgebraDetermining whether a transformation is onto577 views

71) Exploring the solution set of Ax = b  Matrix transformations  Linear AlgebraExploring the solution set of Ax=b (non homogeneous equations)601 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)441 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 matrix438 views

74) Showing that inverses are linear  Matrix transformations  Linear AlgebraShowing that inverse transformations are also linear441 views

75) Deriving a method for determining inverses  Matrix transformations  Linear AlgebraDetermining a method for constructing inverse transformation matrices476 views

76) Example of finding matrix inverse  Matrix transformations  Linear AlgebraExample of calculating the inverse of a matrix468 views

77) Formula for 2x2 inverse  Matrix transformations  Linear AlgebraFiguring out the formula for a 2x2 matrix. Defining the determinant.436 views

78) 3 x 3 determinant  Matrix transformations  Linear AlgebraDeterminants: Finding the determinant of a 3x3 matrix531 views

79) n x n determinant  Matrix transformations  Linear AlgebraDefining the determinant for nxn matrices. An example of a 4x4 determinant.537 views

80) Determinants along other rows/cols  Matrix transformations  Linear AlgebraFinding the determinant by going along other rows or columns434 views

81) Rule of Sarrus of determinants  Matrix transformations  Linear AlgebraA alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus)592 views

82) Determinant when row multiplied by scalar  Matrix transformations  Linear AlgebraThe determinant when a row is multiplied by a scalar663 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 determinant482 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)567 views

85) Duplicate row determinant  Matrix transformations  Linear AlgebraDeterminant of a matrix with duplicate rows553 views

86) Determinant after row operations  Matrix transformations  Linear AlgebraWhat happens to the determinant when we perform a row operation502 views

87) Upper triangular determinant  Matrix transformations  Linear AlgebraThe determinant of an upper triangular matrix468 views

88) Simpler 4x4 determinant  Matrix transformations  Linear AlgebraCalculating a 4x4 determinant by putting in in upper triangular form first.467 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 matrix559 views

90) Determinant as scaling factor  Matrix transformations  Linear AlgebraViewing the determinant of the transformation matrix as a scaling factor of regions484 views

91) Transpose of a matrix  Matrix transformations  Linear AlgebraTranspose of a matrix569 views

92) Determinant of transpose  Matrix transformations  Linear AlgebraProof by induction that transposing a matrix does not change its determinant484 views

93) Transpose of a matrix product  Matrix transformations  Linear AlgebraTaking the transpose of the product of two matrices455 views

94) Transposes of sums and inverses  Matrix transformations  Linear AlgebraTransposes of sums and inverses461 views

95) Transpose of a vector  Matrix transformations  Linear AlgebraTranspose of a column vector. Matrixmatrix products using vectors511 views

96) Rowspace and left nullspace  Matrix transformations  Linear AlgebraRowspace and Left Nullspace601 views

97) Visualizations of left nullspace and rowspace  Linear AlgebraRelationship between left nullspace, rowspace, column space and nullspace.457 views

98) Orthogonal complements  Alternate coordinate systems (bases)  Linear AlgebraOrthogonal Complements as subspaces.506 views

99) rank(a) = rank(transpose of a)  Matrix transformations  Linear AlgebraRank(A) = Rank(transpose of A)413 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) = n456 views

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