In the previous lesson on linear systems, we learned how to use an augmented matrix and the row operations that preserve the relationships in a system to solve a system of linear equations.

At its core, a matrix is a way to represent a table of numbers. Each of the rows and columns in this matrix is represented as a list of numbers. A list of numbers is known as a vector. A row from a matrix is known as a row vector, while a column is known as a column vector.

In this lesson on vectors, we’ll learn more about about column vectors and their associated operations to help us understand certain properties of linear systems. You’ll start by building some geometric intuition of vectors (generally, the word vector refers to the column vector).

Then, we’ll end this lesson by justifying the approach we used in the last lesson to solve the linear system by connecting a few key ideas from matrices and vectors.

As you work through each concept, you’ll get to apply what you’ve learned from within your browser so that there’s no need to use your own machine to do the exercises. The Python environment inside of this course includes answer checking so you can ensure that you’ve fully mastered each concept before learning the next concept.


  • Learn the geometric intution behind vectors.
  • Learn to perform vector operations.
  • Learn the link between linear combinations and solutions to linear systems.

Lesson Outline

  1. From Matrices To Vectors
  2. Geometric Intuition Of Vectors
  3. Vector Operations
  4. Scaling Vectors
  5. Vectors In NumPy
  6. Dot Product
  7. Linear Combination
  8. Linear Combination And Vectors
  9. The Matrix Equation
  10. Next Steps
  11. Takeaways

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