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 \(-(-v)=v\) for every \(v \in V\)

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) \(\quad\left\\{\left(x_{1}, x_{2}, x_{3}\right) \in \mathbf{F}^{3}: x_{1}=5 x_{3}\right\\}\)

Give an example of a nonempty subset \(U\) of \(R^{2}\) 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 \\[ \left.U_{1}+U_{2}=U_{2}+U_{1} ? \text { Is }\left(U_{1}+U_{2}\right)+U_{3}=U_{1}+\left(U_{2}+U_{3}\right) ?\right) \\]