5.1.1 Vectors

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1. Dot product

   1. [E] What’s the geometric interpretation of the dot product of two vectors?
   2. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.

2. Outer product

   1. [E] Given two vectors $a = [3, 2, 1]$ and $b = [-1, 0, 1]$. Calculate the outer product $a^Tb$?
   2. [M] Give an example of how the outer product can be useful in ML.

3. [E] What does it mean for two vectors to be linearly independent?

4. [M] Given two sets of vectors $A = {a_1, a_2, a_3, ..., a_n}$ and $B = {b_1, b_2, b_3, ... , b_m}$. How do you check that they share the same basis?
5. [M] Given $n$ vectors, each of $d$ dimensions. What is the dimension of their span?
6. Norms and metrics
   1. [E] What's a norm? What is $L_0, L_1, L_2, L_{norm}$?
   2. [M] How do norm and metric differ? Given a norm, make a metric. Given a metric, can we make a norm?