Linear algebra
Learn linear algebra—vectors, matrices, transformations, and more.
3 units136 videos33 hours
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Unit 1: Vectors and spaces
Vectors
Vector intro for linear algebra ✓ Check-ins5:49
Real coordinate spaces ✓ Check-ins6:21
Adding vectors algebraically & graphically ✓ Check-ins7:12
Multiplying a vector by a scalar ✓ Check-ins5:44
Vector examples ✓ Check-ins ✎ Fixed a slip25:33
Unit vectors intro ✓ Check-ins ✎ Fixed a slip7:59
Parametric representations of lines ✓ Check-ins ✎ Fixed a slip24:46
Linear combinations and spans
Linear dependence and independence
Subspaces and the basis for a subspace
Vector dot and cross products
Vector dot product and vector length ✓ Check-ins9:10
Proving vector dot product properties ✓ Check-ins10:45
Proof of the Cauchy-Schwarz inequality ✓ Check-ins ✎ Fixed a slip16:55
Vector triangle inequality ✓ Check-ins18:53
Defining the angle between vectors ✓ Check-ins25:11
Defining a plane in R3 with a point and normal vector ✓ Check-ins13:53
Cross product introduction ✓ Check-ins ✎ Fixed a slip15:47
Proof: Relationship between cross product and sin of angle ✓ Check-ins ✎ Fixed a slip18:09
Dot and cross product comparison/intuition ✓ Check-ins19:14
Vector triple product expansion (very optional) ✓ Check-ins ✎ Fixed a slip14:25
Normal vector from plane equation ✓ Check-ins9:58
Point distance to plane ✓ Check-ins ✎ Fixed a slip12:12
Distance between planes ✓ Check-ins ✎ Fixed a slip14:45
Matrices for solving systems by elimination
Null space and column space
Matrix vector products ✓ Check-ins ✎ Fixed a slip21:10
Introduction to the null space of a matrix ✓ Check-ins10:23
Null space 2: Calculating the null space of a matrix ✓ Check-ins ✎ Fixed a slip13:07
Null space 3: Relation to linear independence ✓ Check-ins9:32
Column space of a matrix ✓ Check-ins10:40
Null space and column space basis ✓ Check-ins25:13
Visualizing a column space as a plane in R3 ✓ Check-ins21:11
Proof: Any subspace basis has same number of elements ✓ Check-ins ✎ Fixed a slip21:35
Dimension of the null space or nullity ✓ Check-ins13:59
Dimension of the column space or rank ✓ Check-ins ✎ Fixed a slip12:48
Showing relation between basis cols and pivot cols ✓ Check-ins8:33
Showing that the candidate basis does span C(A) ✓ Check-ins ✎ Fixed a slip13:40
Unit 2: Matrix transformations
Functions and linear transformations
A more formal understanding of functions ✓ Check-ins16:01
Vector transformations ✓ Check-ins ✎ Fixed a slip14:19
Linear transformations ✓ Check-ins13:52
Matrix vector products as linear transformations ✓ Check-ins17:04
Linear transformations as matrix vector products ✓ Check-ins17:32
Image of a subset under a transformation ✓ Check-ins ✎ Fixed a slip18:11
im(T): Image of a transformation ✓ Check-ins ✎ Fixed a slip16:37
Preimage of a set ✓ Check-ins ✎ Fixed a slip5:23
Preimage and kernel example ✓ Check-ins15:23
Sums and scalar multiples of linear transformations ✓ Check-ins ✎ Fixed a slip15:09
More on matrix addition and scalar multiplication ✓ Check-ins ✎ Fixed a slip10:41
Linear transformation examples
Linear transformation examples: Scaling and reflections ✓ Check-ins ✎ Fixed a slip15:13
Linear transformation examples: Rotations in R2 ✓ Check-ins ✎ Fixed a slip17:52
Rotation in R3 around the x-axis ✓ Check-ins12:18
Unit vectors ✓ Check-ins6:59
Introduction to projections ✓ Check-ins14:37
Expressing a projection on to a line as a matrix vector prod ✓ Check-ins ✎ Fixed a slip16:40
Transformations and matrix multiplication
Compositions of linear transformations 1 ✓ Check-ins ✎ Fixed a slip12:21
Compositions of linear transformations 2 ✓ Check-ins ✎ Fixed a slip16:30
Matrix product examples ✓ Check-ins18:14
Matrix product associativity ✓ Check-ins11:59
Distributive property of matrix products ✓ Check-ins ✎ Fixed a slip9:52
Inverse functions and transformations
Introduction to the inverse of a function ✓ Check-ins ✎ Fixed a slip18:53
Proof: Invertibility implies a unique solution to f(x)=y ✓ Check-ins ✎ Fixed a slip22:41
Surjective (onto) and injective (one-to-one) functions ✓ Check-ins9:31
Relating invertibility to being onto and one-to-one ✓ Check-ins6:31
Determining whether a transformation is onto ✓ Check-ins ✎ Fixed a slip25:51
Exploring the solution set of Ax = b ✓ Check-ins ✎ Fixed a slip16:34
Matrix condition for one-to-one transformation ✓ Check-ins19:59
Simplifying conditions for invertibility ✓ Check-ins6:37
Showing that inverses are linear ✓ Check-ins21:25
Finding inverses and determinants
Deriving a method for determining inverses ✓ Check-ins ✎ Fixed a slip18:00
Example of finding matrix inverse ✓ Check-ins6:22
Formula for 2x2 inverse ✓ Check-ins18:20
3 x 3 determinant ✓ Check-ins10:01
n x n determinant ✓ Check-ins18:40
Determinants along other rows/cols ✓ Check-ins ✎ Fixed a slip9:03
Rule of Sarrus of determinants ✓ Check-ins ✎ Fixed a slip7:18
More determinant depth
Determinant when row multiplied by scalar ✓ Check-ins ✎ Fixed a slip13:20
(correction) scalar multiplication of row ✓ Check-ins2:52
Determinant when row is added ✓ Check-ins16:55
Duplicate row determinant ✓ Check-ins8:19
Determinant after row operations ✓ Check-ins10:25
Upper triangular determinant ✓ Check-ins ✎ Fixed a slip8:07
Simpler 4x4 determinant ✓ Check-ins ✎ Fixed a slip9:13
Determinant and area of a parallelogram ✓ Check-ins21:37
Determinant as scaling factor ✓ Check-ins20:09
Transpose of a matrix
Transpose of a matrix ✓ Check-ins8:37
Determinant of transpose ✓ Check-ins ✎ Fixed a slip14:10
Transpose of a matrix product ✓ Check-ins ✎ Fixed a slip8:50
Transposes of sums and inverses ✓ Check-ins8:41
Transpose of a vector ✓ Check-ins ✎ Fixed a slip12:06
Rowspace and left nullspace ✓ Check-ins ✎ Fixed a slip23:18
Visualizations of left nullspace and rowspace ✓ Check-ins ✎ Fixed a slip19:50
rank(a) = rank(transpose of a) ✓ Check-ins11:13
Showing that A-transpose x A is invertible ✓ Check-ins ✎ Fixed a slip12:34
Unit 3: Alternate coordinate systems (bases)
Orthogonal complements
Orthogonal complements ✓ Check-ins ✎ Fixed a slip22:08
dim(v) + dim(orthogonal complement of v) = n ✓ Check-ins9:27
Representing vectors in rn using subspace members ✓ Check-ins ✎ Fixed a slip27:00
Orthogonal complement of the orthogonal complement ✓ Check-ins ✎ Fixed a slip12:18
Orthogonal complement of the nullspace ✓ Check-ins3:27
Unique rowspace solution to Ax = b ✓ Check-ins19:12
Rowspace solution to Ax = b example ✓ Check-ins ✎ Fixed a slip19:38
Orthogonal projections
Projections onto subspaces ✓ Check-ins17:26
Visualizing a projection onto a plane ✓ Check-ins9:28
A projection onto a subspace is a linear transformation ✓ Check-ins16:15
Subspace projection matrix example ✓ Check-ins ✎ Fixed a slip13:04
Another example of a projection matrix ✓ Check-ins21:35
Projection is closest vector in subspace ✓ Check-ins9:05
Least squares approximation ✓ Check-ins15:32
Least squares examples ✓ Check-ins18:50
Another least squares example ✓ Check-ins ✎ Fixed a slip13:25
Change of basis
Coordinates with respect to a basis ✓ Check-ins ✎ Fixed a slip16:08
Change of basis matrix ✓ Check-ins17:55
Invertible change of basis matrix ✓ Check-ins13:34
Transformation matrix with respect to a basis ✓ Check-ins18:02
Alternate basis transformation matrix example ✓ Check-ins13:20
Alternate basis transformation matrix example part 2 ✓ Check-ins12:36
Changing coordinate systems to help find a transformation matrix ✓ Check-ins ✎ Fixed a slip28:59
Orthonormal bases and the Gram-Schmidt process
Introduction to orthonormal bases ✓ Check-ins11:16
Coordinates with respect to orthonormal bases ✓ Check-ins15:28
Projections onto subspaces with orthonormal bases ✓ Check-ins16:14
Finding projection onto subspace with orthonormal basis example ✓ Check-ins6:42
Example using orthogonal change-of-basis matrix to find transformation matrix ✓ Check-ins ✎ Fixed a slip27:04
Orthogonal matrices preserve angles and lengths ✓ Check-ins11:16
The Gram-Schmidt process ✓ Check-ins19:24
Gram-Schmidt process example ✓ Check-ins ✎ Fixed a slip13:14
Gram-Schmidt example with 3 basis vectors ✓ Check-ins13:57
Eigen-everything
Introduction to eigenvalues and eigenvectors ✓ Check-ins7:43
Proof of formula for determining eigenvalues ✓ Check-ins9:19
Example solving for the eigenvalues of a 2x2 matrix ✓ Check-ins5:39
Finding eigenvectors and eigenspaces example ✓ Check-ins14:34
Eigenvalues of a 3x3 matrix ✓ Check-ins ✎ Fixed a slip14:08
Eigenvectors and eigenspaces for a 3x3 matrix ✓ Check-ins ✎ Fixed a slip15:34
Showing that an eigenbasis makes for good coordinate systems ✓ Check-ins13:09
Learn this with a class
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