Linear Algebra Foundations
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Linear algebra, starting from the very first vector.

Short lessons, one idea at a time. Every question is checked step by step, and when something goes wrong you are told exactly which number to look at again. No account, no timer.

Start learning Browse skills
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u plus v equals the vector with components 3, 1 plus the vector with components 1, 2 equals the vector with components 4, 3
Two arrows, one sum. You will be able to do this by lesson 2.

The path

  1. Lesson 1: What is a vector? A vector is an ordered list of numbers. Learn to read its components and draw it as an arrow.
  2. Lesson 2: Add and subtract vectors Combine two vectors by adding or subtracting their matching components.
  3. Lesson 3: Multiply a vector by a number Stretch, shrink or flip a vector by multiplying every component by a scalar.
  4. Lesson 4: What is a matrix? A matrix is a rectangle of numbers. Learn rows, columns, size and how to name one entry.
  5. Lesson 5: Add and subtract matrices Add or subtract two matrices of the same size, one position at a time.
  6. Lesson 6: Multiply a matrix by a number Multiply every entry of a matrix by the same scalar.
  7. Lesson 7: Transpose a matrix Swap rows and columns, and see why the size of a matrix can change.
  8. Lesson 8: Dot product basics Multiply matching components and add: two vectors in, one number out.

How each lesson works

  1. Understand

    A short explanation and a picture, in plain English.

  2. See it worked

    One example, solved one step at a time.

  3. Try, check, continue

    One question at a time, with hints when you want them and feedback that points to the exact mistake.