Linear algebra

About Course

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. Vectors. We will begin our journey through linear algebra by defining and conceptualizing what a vector is (rather than starting with matrices and matrix operations ... This is the property of khanacademy.org and is available for free on their website.

Topics for this course

155 Lessons

Vectors and spaces

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

Alternate coordinate systems (bases)

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