Chapter 1 Suggested Problems

Suggested Problems

  1. Show that a vector space over \mathbf{C} of dimension n can be thought of as a vector space over \mathbf{R} of dimension 2n.

  2. Let V be the set of infinite sequences (a_1,a_2,\ldots) where a_{i}\in\mathbf{R} for all i and all but finitely many of the a_{i} are zero. Show that V is a real vector space and that the set of vectors e_{i} having a one in position i and zeroes elsewhere (for all i=1,2,\ldots) is a basis for V.

  3. Look at the proof of the result that every vector space has a basis. We established this by using Zorn’s lemma to prove that there is a maximal linearly independent set, and that this set is a basis. You can instead consider spanning sets, which are partially ordered in the “opposite” direction. Can you use Zorn’s lemma to show that there is a minimal spanning set, and that this must be a basis? The argument would be that a descending chain of spanning sets has a lower bound, so there’s a minimal spanning set, which must be a basis. Does this work?

  4. Study each of the following system of equations and describe their solutions.

  1. \begin{aligned} x + y + z &= 6\\ 2x - y + z &= 3\\ x + 2y - z &= 2 \end{aligned}

  2. \begin{aligned} x + y + z &= 6\\ x + 2y + z &= 9\\ 5x + 8y + 5z &= 39 \end{aligned}

  3. \begin{aligned} x + y + z &= 1\\ x + z &= 3\\ x - y + z &= 0 \end{aligned}