Free Course Image Linear Algebra Essentials

Free online courseLinear Algebra Essentials

Duration of the online course: 3 hours and 0 minutes

New

Free linear algebra course with visual intuition: vectors, matrices, transformations, determinants, eigenvalues, change of basis, and vector spaces.

In this free course, learn about

  • Vectors and Linear Combinations
  • Matrix Operations and Transformations
  • Inverse Matrices and Dimensional Mappings
  • Cross Products and Geometric Interpretation
  • Coordinate Systems and Eigenvalues
  • Abstract Vector Spaces and Linearity

Course Description

Linear Algebra Essentials is a free online course in the School Subjects category, within Algebra, designed to build strong intuition for the ideas that power modern math, science, and computing. It guides you from the meaning of vectors and linear combinations to the geometric heart of span and basis, helping you see why these concepts matter rather than just how to compute them.

You will develop a clear understanding of linear transformations and how matrices represent them, including how composition naturally leads to matrix multiplication. The course highlights three-dimensional transformations and uses visuals to make determinants feel like a measure of scaling and orientation, connecting computation to geometry.

As you progress, you will explore inverse matrices, column space, and null space, along with the role of nonsquare matrices as dimension-changing transformations. The course also explains dot products through the lens of duality, then connects cross products to linear transformations to deepen conceptual clarity.

Later sections tie key problem-solving tools to geometric meaning, including Cramer’s rule and change of basis. You will also gain a practical, intuitive grasp of eigenvectors and eigenvalues, including a helpful shortcut for finding eigenvalues, and finish with a perspective on abstract vector spaces that unifies the ideas you have learned.

If you want a visually driven, intuition-first introduction that makes linear algebra click, this course provides a coherent path from fundamentals to the big-picture structure behind the calculations.

Course content

  • Video class: Vectors | Chapter 1, Essence of linear algebra 09m
  • Exercise: How do you compute the sum of two vectors using their coordinate lists?
  • Video class: Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra 09m
  • Exercise: What does it mean for a set of vectors to be a basis of a space (technical definition)?
  • Video class: Linear transformations and matrices | Chapter 3, Essence of linear algebra 10m
  • Exercise: In a 2D linear transformation represented by a 2×2 matrix, what do the matrix columns represent?
  • Video class: Matrix multiplication as composition | Chapter 4, Essence of linear algebra 10m
  • Exercise: What does multiplying two matrices represent geometrically?
  • Video class: Three-dimensional linear transformations | Chapter 5, Essence of linear algebra 04m
  • Exercise: In 3D, how do you build the 3×3 matrix that represents a linear transformation?
  • Video class: The determinant | Chapter 6, Essence of linear algebra 10m
  • Exercise: What does the determinant of a 2D linear transformation represent?
  • Video class: Inverse matrices, column space and null space | Chapter 7, Essence of linear algebra 12m
  • Exercise: What does the rank of a matrix represent geometrically?
  • Video class: Nonsquare matrices as transformations between dimensions | Chapter 8, Essence of linear algebra 04m
  • Exercise: What is the geometric meaning of a 3×2 matrix used as a linear transformation?
  • Video class: Dot products and duality | Chapter 9, Essence of linear algebra 14m
  • Exercise: What geometric meaning is emphasized for the dot product v · w?
  • Video class: Cross products | Chapter 10, Essence of linear algebra 08m
  • Exercise: In 2D, how can you compute the (signed) area associated with v × w?
  • Video class: Cross products in the light of linear transformations | Chapter 11, Essence of linear algebra 13m
  • Exercise: What geometric properties characterize the 3D cross product v × w?
  • Video class: Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra 12m
  • Exercise: In Cramer's rule (2D case), how do you compute the x-coordinate of the solution vector?
  • Video class: Change of basis | Chapter 13, Essence of linear algebra 12m
  • Exercise: How do you translate a vector from one basis to another using a change-of-basis matrix?
  • Video class: Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra 17m
  • Exercise: What condition must be true for a real number λ to be an eigenvalue of a matrix A?
  • Video class: A quick trick for computing eigenvalues | Chapter 15, Essence of linear algebra 13m
  • Exercise: For a 2×2 matrix, which pair of quantities lets you quickly write the eigenvalues as m ± √(m² − p)?
  • Video class: Abstract vector spaces | Chapter 16, Essence of linear algebra 16m
  • Exercise: Which two properties define a linear transformation (including for functions like the derivative)?

This free course includes:

3 hours and 0 minutes of online video course

Digital certificate of course completion (Free)

Exercises to train your knowledge

100% free, from content to certificate

Ready to get started?Download the app and get started today.

Install the app now

to access the course
Icon representing technology and business courses

Over 5,000 free courses

Programming, English, Digital Marketing and much more! Learn whatever you want, for free.

Calendar icon with target representing study planning

Study plan with AI

Our app's Artificial Intelligence can create a study schedule for the course you choose.

Professional icon representing career and business

From zero to professional success

Improve your resume with our free Certificate and then use our Artificial Intelligence to find your dream job.

You can also use the QR Code or the links below.

QR Code - Download Cursa - Online Courses

More free courses at Algebra

Free Ebook + Audiobooks! Learn by listening or reading!

Download the App now to have access to + 5000 free courses, exercises, certificates and lots of content without paying anything!

  • 100% free online courses from start to finish

    Thousands of online courses in video, ebooks and audiobooks.

  • More than 60 thousand free exercises

    To test your knowledge during online courses

  • Valid free Digital Certificate with QR Code

    Generated directly from your cell phone's photo gallery and sent to your email

Cursa app on the ebook screen, the video course screen and the course exercises screen, plus the course completion certificate