Exercise 22

We can show that every basis for $\mathbb{R}^{3}$ contains the same number of vectors.

• Show that no linearly independent subset of $\mathbb{R}^{3}$ contains more than three vectors.

• Show that no spanning subset of $\mathbb{R}^{3}$ contains fewer than three vectors. Hint: recall how to calculate the span of a set and show that this method cannot yield all of $\mathbb{R}^{3}$ when we apply it to fewer than three vectors.