Vectors and the dot product

After this lesson you can

• Move between the arrow picture and the list-of-numbers picture of a vector
• Explain linear combinations and span, and recognise when two vectors are dependent
• Compute ${\ell }_1$, ${\ell }_2$ and ${\ell }_{\infty }$ norms and say what each one measures
• Derive $\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$ from the law of cosines
• Compute projections and cosine similarity, and read them in attention formulas

In this lesson

1. One object, two pictures
2. Adding and stretching
3. Linear combinations and span
4. How long is a vector? Norms
5. The dot product, as arithmetic
6. The dot product, as geometry
7. In code
8. Where this goes: attention