Questions

01

Subspaces

Suppose a set $W$ satisfies the symbolic hypothesis above:

$${\mathbf{w}}_1,{\mathbf{w}}_2\in W\text{implies}a{\mathbf{w}}_1+b{\mathbf{w}}_2\in W,\text{any }a,b.$$

Show that any linear combination of $n$ vectors from $W$ must lie in $W$.

02

Iterating from $2$ to $n$

How many distinct subspaces of ${\mathbb{R}}^n$ can you describe using the general idea of the previous example?