91Ó°ÊÓ

Suppose \(a\) and \(b\) are real numbers, not both \(0 .\) Find real numbers \(c\) and \(d\) such that $$ 1 /(a+b i)=c+d i $$

Show that $$ \frac{-1+\sqrt{3} i}{2} $$ is a cube root of 1 (meaning that its cube equals 1).

Prove that if \(a \in \mathbf{F}, v \in V\), and \(a v=0\), then \(a=0\) or \(v=0\).

For each of the following subsets of \(\mathbf{F}^{3}\), determine whether it is a subspace of \(\mathbf{F}^{3}\) : (a) \(\quad\left\\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1}+2 x_{2}+3 x_{3}=0\right\\}\); (b) \(\quad\left\\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1}+2 x_{2}+3 x_{3}=4\right\\}\); (c) \(\quad\left\\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1} x_{2} x_{3}=0\right\\}\); (d) \(\left\\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1}=5 x_{3}\right\\}\).

Give an example of a nonempty subset \(U\) of \(\mathbf{R}^{2}\) such that \(U\) is closed under addition and under taking additive inverses (meaning \(-u \in U\) whenever \(u \in U)\), but \(U\) is not a subspace of \(\mathbf{R}^{2}\).

Give an example of a nonempty subset \(U\) of \(\mathbf{R}^{2}\) such that \(U\) is closed under scalar multiplication, but \(U\) is not a subspace of \(\mathbf{R}^{2}\).

Prove that the intersection of any collection of subspaces of \(V\) is a subspace of \(V\).

Suppose that \(U\) is a subspace of \(V\). What is \(U+U ?\)

Is the operation of addition on the subspaces of \(V\) commutative? Associative? (In other words, if \(U_{1}, U_{2}, U_{3}\) are subspaces of \(V\), is \(U_{1}+U_{2}=U_{2}+U_{1} ?\) Is \(\left.\left(U_{1}+U_{2}\right)+U_{3}=U_{1}+\left(U_{2}+U_{3}\right) ?\right)\)

Does the operation of addition on the subspaces of \(V\) have an additive identity? Which subspaces have additive inverses?