Homework 3: Vector Spaces

1. Section 4.1, #2, 7, TF (True-False Exercises).
2. Let \(V\) be the set of all matrices of real numbers with one column and two rows, with addition and scalar multiplication defined as follows: $$ \btwo{u_1}{u_2}+\btwo{v_1}{v_2}=\btwo{u_1+v_1}{u_2v_2}\qquad \text{and}\qquad c \btwo{u_1}{u_2}=\btwo{cu_1}{u_2} $$ With this addition and scalar multiplication, \(V\) is not a vector space. Identify all vector space axioms which fail, and briefly explain why each of these axioms fail (in most cases, a counterexample will be sufficient). If axiom 3 holds, then give an explicit additive identity.
3. Let \(V\) be a vector space. Let \(\bu,\bv\) and \(\bw\) be vectors in \(V\), and let \(b\) and \(c\) be scalars. Using only the definition of a vector space, prove
   1. \(c\,\bz=\bz\)
   2. \((\bu+\bv)+\bw=\bv+(\bw+\bu)\) (Hint: use only the commutative and associative axioms.)
   3. \((b+c)(\bu+\bv)=(c\bu + c\bv) + (b\bu+b\bv)\)
4. (Challenge) Prove that there does not exist a vector space which contains exactly two elements.