Exercise 3

Remember that the outer product of two vectors $\uvec$ and $\vvec$ is the matrix $\uvec~\vvec^T$.

1. Suppose that $\uvec = \twovec{2}{-3}$ and $\vvec=\threevec201$. Evaluate the outer product $\uvec~\vvec^T$. To get a clearer sense of how this works, perform this operation without using technology.

   How is each of the columns of $\uvec~\vvec^T$ related to $\uvec$?

2. Suppose $\uvec$ and $\vvec$ are general vectors. What is $\rank(\uvec~\vvec^T)$ and what is a basis for its column space $\col(\uvec~\vvec^T)$?

3. Suppose that $\uvec$ is a unit vector. What is the effect of multiplying a vector by the matrix $\uvec~\uvec^T$?