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Linear algebra

#### Course content

#### Vectors and spaces

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#### Matrix transformations

#### Alternate coordinate systems (bases)

Free

155 Lessons

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)