Linear algebra | 1 Big School (By Blue Brain)

Course Content

Vectors and spaces

Vectors
Vector intro for linear algebra
Real coordinate spaces
Adding vectors algebraically & graphically
Multiplying a vector by a scalar
Vector examples
Unit vectors intro
Parametric representations of lines
Linear combinations and spans
Linear combinations and span
Linear dependence and independence
Introduction to linear independence
More on linear independence
Span and linear independence example
Subspaces and the basis for a subspace
Linear subspaces
Basis of a subspace
Vector dot and cross products
Vector dot product and vector length
Proving vector dot product properties
Proof of the Cauchy-Schwarz inequality
Vector triangle inequality
Defining the angle between vectors
Defining a plane in R3 with a point and normal vector
Cross product introduction
Proof: Relationship between cross product and sin of angle
Dot and cross product comparison/intuition
Vector triple product expansion (very optional)
Normal vector from plane equation
Point distance to plane
Distance between planes
Matrices for solving systems by elimination
Solving a system of 3 equations and 4 variables using matrix row-echelon form
Solving linear systems with matrices
Using matrix row-echelon form in order to show a linear system has no solutions
Null space and column space
Matrix vector products
Introduction to the null space of a matrix
Null space 2: Calculating the null space of a matrix
Null space 3: Relation to linear independence
Column space of a matrix
Null space and column space basis
Visualizing a column space as a plane in R3
Proof: Any subspace basis has same number of elements
Dimension of the null space or nullity
Dimension of the column space or rank
Showing relation between basis cols and pivot cols
Showing that the candidate basis does span C(A)

Matrix transformations

Functions and linear transformations
A more formal understanding of functions
Vector transformations
Linear transformations
Matrix vector products as linear transformations
Linear transformations as matrix vector products
Image of a subset under a transformation
im(T): Image of a transformation
Preimage of a set
Preimage and kernel example
Sums and scalar multiples of linear transformations
More on matrix addition and scalar multiplication
Linear transformation examples
Linear transformation examples: Scaling and reflections
Linear transformation examples: Rotations in R2
Rotation in R3 around the x-axis
Unit vectors
Introduction to projections
Expressing a projection on to a line as a matrix vector prod
Transformations and matrix multiplication
Compositions of linear transformations 1
Compositions of linear transformations 2
Matrix product examples
Matrix product associativity
Distributive property of matrix products
Inverse functions and transformations
Introduction to the inverse of a function
Proof: Invertibility implies a unique solution to f(x)=y
Surjective (onto) and injective (one-to-one) functions
Relating invertibility to being onto and one-to-one
Determining whether a transformation is onto
Exploring the solution set of Ax = b
Matrix condition for one-to-one transformation
Simplifying conditions for invertibility
Showing that inverses are linear
Finding inverses and determinants
Deriving a method for determining inverses
Example of finding matrix inverse
Formula for 2×2 inverse
3 x 3 determinant
n x n determinant
Determinants along other rows/cols
Rule of Sarrus of determinants
More determinant depth
Determinant when row multiplied by scalar
(correction) scalar multiplication of row
Determinant when row is added
Duplicate row determinant
Determinant after row operations
Upper triangular determinant
Simpler 4×4 determinant
Determinant and area of a parallelogram
Determinant as scaling factor
Transpose of a matrix
Transpose of a matrix
Determinant of transpose
Transpose of a matrix product
Transposes of sums and inverses
Transpose of a vector
Rowspace and left nullspace
Visualizations of left nullspace and rowspace
rank(a) = rank(transpose of a)
Showing that A-transpose x A is invertible
Orthogonal complements
Orthogonal complements
dim(v) + dim(orthogonal complement of v) = n
Representing vectors in rn using subspace members
Orthogonal complement of the orthogonal complement
Orthogonal complement of the nullspace
Unique rowspace solution to Ax = b
Rowspace solution to Ax = b example

Alternate coordinate systems (bases)

Orthogonal projections
Projections onto subspaces
Visualizing a projection onto a plane
A projection onto a subspace is a linear transformation
Subspace projection matrix example
Another example of a projection matrix
Projection is closest vector in subspace
Least squares approximation
Least squares examples
Another least squares example
Change of basis
Coordinates with respect to a basis
Change of basis matrix
Invertible change of basis matrix
Transformation matrix with respect to a basis
Alternate basis transformation matrix example
Alternate basis transformation matrix example part 2
Changing coordinate systems to help find a transformation matrix
Orthonormal bases and the Gram-Schmidt process
Introduction to orthonormal bases
Coordinates with respect to orthonormal bases
Projections onto subspaces with orthonormal bases
Finding projection onto subspace with orthonormal basis example
Example using orthogonal change-of-basis matrix to find transformation matrix
Orthogonal matrices preserve angles and lengths
The Gram-Schmidt process
Gram-Schmidt process example
Gram-Schmidt example with 3 basis vectors
Eigen-everything
Introduction to eigenvalues and eigenvectors
Proof of formula for determining eigenvalues
Example solving for the eigenvalues of a 2×2 matrix
Finding eigenvectors and eigenspaces example
Eigenvalues of a 3×3 matrix
Eigenvectors and eigenspaces for a 3×3 matrix
Showing that an eigenbasis makes for good coordinate systems

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Course Content

Vectors and spaces

Vectors

Vector intro for linear algebra

Real coordinate spaces

Adding vectors algebraically & graphically

Multiplying a vector by a scalar

Vector examples

Unit vectors intro

Parametric representations of lines

Linear combinations and spans

Linear combinations and span

Linear dependence and independence

Introduction to linear independence

More on linear independence

Span and linear independence example

Subspaces and the basis for a subspace

Linear subspaces

Basis of a subspace

Vector dot and cross products

Vector dot product and vector length

Proving vector dot product properties

Proof of the Cauchy-Schwarz inequality

Vector triangle inequality

Defining the angle between vectors

Defining a plane in R3 with a point and normal vector

Cross product introduction

Proof: Relationship between cross product and sin of angle

Dot and cross product comparison/intuition

Vector triple product expansion (very optional)

Normal vector from plane equation

Point distance to plane

Distance between planes

Matrices for solving systems by elimination

Solving a system of 3 equations and 4 variables using matrix row-echelon form

Solving linear systems with matrices

Using matrix row-echelon form in order to show a linear system has no solutions

Null space and column space

Matrix vector products

Introduction to the null space of a matrix

Null space 2: Calculating the null space of a matrix

Null space 3: Relation to linear independence

Column space of a matrix

Null space and column space basis

Visualizing a column space as a plane in R3

Proof: Any subspace basis has same number of elements

Dimension of the null space or nullity

Dimension of the column space or rank

Showing relation between basis cols and pivot cols

Showing that the candidate basis does span C(A)

Matrix transformations

Functions and linear transformations

A more formal understanding of functions

Vector transformations

Linear transformations

Matrix vector products as linear transformations

Linear transformations as matrix vector products

Image of a subset under a transformation

im(T): Image of a transformation

Preimage of a set

Preimage and kernel example

Sums and scalar multiples of linear transformations

More on matrix addition and scalar multiplication

Linear transformation examples

Linear transformation examples: Scaling and reflections

Linear transformation examples: Rotations in R2

Rotation in R3 around the x-axis

Unit vectors

Introduction to projections

Expressing a projection on to a line as a matrix vector prod

Transformations and matrix multiplication

Compositions of linear transformations 1

Compositions of linear transformations 2

Matrix product examples

Matrix product associativity

Distributive property of matrix products

Inverse functions and transformations

Introduction to the inverse of a function

Proof: Invertibility implies a unique solution to f(x)=y

Surjective (onto) and injective (one-to-one) functions