Exercise 26

A vector space over the complex numbers $\mathbb{C}$ has the same definition as a vector space over the reals except that scalars are drawn from $\mathbb{C}$ instead of from $\mathbb{R}$. Show that each of these is a vector space over the complex numbers. (Recall how complex numbers add and multiply: $(a_{0}+a_{1}i)+(b_{0}+b_{1}i)=(a_{0}+b_{0})+(a_{1}+b_{1})i$ and $(a_{0}+a_{1}i)(b_{0}+b_{1}i)=(a_{0}b_{0}-a_{1}b_{1})+(a_{0}b_{1}+a_{1}b_{0})i$.)

• The set of degree two polynomials with complex coefficients

• This set

  $\{\hskip 1.00006pt\begin{pmatrix}0&a\\ b&0\end{pmatrix}\mid a,b\in\mathbb{C}\text{\ and\ }a+b=0+0i\hskip 1.00006pt\}$