Linear Algebra

90 votes
Free Closed [?]
Linear Algebra

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.
Introduction to matrices. Matrix multiplication (part 1). Matrix multiplication (part 2). Idea Behind Inverting a 2x2 Matrix. Inverting matrices (part 2). Inverting Matrices (part 3). Matrices to solve a system of equations. Matrices to solve a vector combination problem. Singular Matrices. 3-variable linear equations (part 1). Solving 3 Equations with 3 Unknowns. Introduction to Vectors. Vector Examples. Parametric Representations of Lines. Linear Combinations and Span. Introduction to Linear Independence. More on linear independence. Span and Linear Independence Example. Linear Subspaces. Basis of a Subspace. 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. Matrices: Reduced Row Echelon Form 1. Matrices: Reduced Row Echelon Form 2. Matrices: Reduced Row Echelon Form 3. 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). 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: 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. Compositions of Linear Transformations 1. Compositions of Linear Transformations 2. Matrix Product Examples. Matrix Product Associativity. Distributive Property of Matrix Products. 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 trans. Simplifying conditions for invertibility. Showing that Inverses are Linear. Deriving a method for determining inverses. Example of Finding Matrix Inverse. Formula for 2x2 inverse. 3x3 Determinant. nxn Determinant. Determinants along other rows/cols. Rule of Sarrus of Determinants. 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 4x4 determinant. Determinant and area of a parallelogram. Determinant as Scaling Factor. 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. Orthogonal Complements. Rank(A) = Rank(transpose of A). 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. Showing that A-transpose x A is invertible. Projections onto Subspaces. Visualizing a projection onto a plane. A Projection onto a Subspace is a Linear Transforma. 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. 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. 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. Introduction to Eigenvalues and Eigenvectors. Proof of formula for determining Eigenvalues. Example solving for the eigenvalues of a 2x2 matrix. Finding Eigenvectors and Eigenspaces example. Eigenvalues of a 3x3 matrix. Eigenvectors and Eigenspaces for a 3x3 matrix. Showing that an eigenbasis makes for good coordinate systems. Vector Triple Product Expansion (very optional). Normal vector from plane equation. Point distance to plane. Distance Between Planes.

Categories: Mathematics

Comments

Alternatives

10 0 0

This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines such as physics, economics and…

5 0 0

This course is an introduction to linear algebra.  It has been argued that linear algebra constitutes half of all mathematics. …

Prerequisites

-- no prerequsites found for the course --
If you can suggest any prerequisite, please let us know.

Paths

  • Applied Mathematics

    Applied Mathematics degree

  • Mathematics

    This is the path to get alternative degree in Mathematics with MOOCs and other online courses



See all Education Paths

Certification Exams

-- there are no exams to get certification after this course --
If your company does certification for those who completed this course then register your company as certification vendor and add your exams to the Exams Directory.

Students

People who learned this course or plan to learn. These people added the course to their Education Passport or their Personal Education Path.

Alberto Mesa Mohsin Zafar Varoon Silochan Ольга Ларькина Рустем Жумагулов

If you learned this course or plan to learn then add it to your Personal Education Path or your Education Passport

Similar courses

Courses related to the course subject

90 0 0

Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an a…

56 0 0

We explore creating and moving between various coordinate systems. Orthogonal Complements. dim(V) + dim(orthogonal complement of V…

56 0 0

Understanding how we can map one set of vectors to another set. Matrices used to define linear transformations. A more formal und…

40 0 0

Let's get our feet wet by thinking in terms of vectors and spaces. Introduction to Vectors. Vector Examples. Scaling vectors.

5 0 0

This course is an introduction to linear algebra.  It has been argued that linear algebra constitutes half of all mathematics. …

13 0 0

Advanced Analytic Methods in Science and Engineering is a comprehensive treatment of the advanced methods of applied mathematics.

10 0 0

This course analyzes the functions of a complex variable and the calculus of residues. It also covers subjects such as ordinary di…

8 0 0

This graduate-level course focuses on current research topics in computational complexity theory. Topics include: Nondeterministic…

The topics for this course cover various aspects of complexity theory, such as  the basic time and space classes, the po…

16 0 0

The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various…

Let us know when you did the course Linear Algebra.

Started on: Completed on:
Your grade (if any):
Comments:

Add the course Linear Algebra to My Personal Education Path.

Start the course on:
Duration of study:
Notes:

Successfully added to your path.

View your path

Select what exam to connect to the course. The course will be displayed on the exam page in the list of courses supported for certification with the exam.


Notes about how the exam certifies students of the course (optional):