Definition of Vector Space

Prove that $-(-v) = v$ for every $v \in V$.

Suppose $a \in \mathbf{F}, v \in V$, and $a v = 0$. Prove that $a = 0$ or $v = 0$.

Suppose $v, w \in V$. Explain why there exists a unique $x \in V$ such that $v+3 x = w$.

The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space (1.20). Which one?

Definition 1.20 (vector space): a vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold. Commutativity: $u+v = v+u$ for all $u, v \in V$. Associativity: $(u+v)+w = u+(v+w)$ and $(a b) v = a(b v)$ for all $u, v, w \in V$ and for all $a, b \in \mathbf{F}$. Additive identity: there exists an element $0 \in V$ such that $v+0 = v$ for all $v \in V$. Additive inverse: for every $v \in V$, there exists $w \in V$ such that $v+w = 0$. Multiplicative identity: $1 v = v$ for all $v \in V$. Distributive properties: $a(u+v) = a u+a v$ and $(a+b) v = a v+b v$ for all $a, b \in \mathbf{F}$ and all $u, v \in V$.

Show that in the definition of a vector space (1.20), the additive inverse condition can be replaced with the condition that

Here the 0 on the left side is the number 0, and the 0 on the right side is the additive identity of $V$.

Definition 1.20 (vector space): a vector space is a set $V$ along with an addition on $V$ and a scalar multiplication on $V$ such that the following properties hold. Commutativity: $u+v = v+u$ for all $u, v \in V$. Associativity: $(u+v)+w = u+(v+w)$ and $(a b) v = a(b v)$ for all $u, v, w \in V$ and for all $a, b \in \mathbf{F}$. Additive identity: there exists an element $0 \in V$ such that $v+0 = v$ for all $v \in V$. Additive inverse: for every $v \in V$, there exists $w \in V$ such that $v+w = 0$. Multiplicative identity: $1 v = v$ for all $v \in V$. Distributive properties: $a(u+v) = a u+a v$ and $(a+b) v = a v+b v$ for all $a, b \in \mathbf{F}$ and all $u, v \in V$.

Let $\infty$ and $-\infty$ denote two distinct objects, neither of which is in $\mathbb {R}$. Define an addition and scalar multiplication on $\mathbb {R} \cup \left\{ \infty ,-\infty \right\}$ as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and for $t \in \mathbb {R}$ define

and

With these operations of addition and scalar multiplication, is $\mathbb {R} \cup \left\{ \infty ,-\infty \right\}$ a vector space over $\mathbb {R}$? Explain.

Suppose $S$ is a nonempty set. Let $V^{S}$ denote the set of functions from $S$ to $V$. Define a natural addition and scalar multiplication on $V^{S}$, and show that $V^{S}$ is a vector space with these definitions.

Suppose $V$ is a real vector space.

• The complexification of $V$, denoted by $V_{\mathrm{C}}$, equals $V \times V$. An element of $V_{\mathrm{C}}$ is an ordered pair $(u, v)$, where $u, v \in V$, but we write this as $u+i v$.

• Addition on $V_{\mathrm{C}}$ is defined by

  $$\left(u_{1}+i v_{1}\right)+\left(u_{2}+i v_{2}\right) = \left(u_{1}+u_{2}\right)+i\left(v_{1}+v_{2}\right)$$

  for all $u_{1}, v_{1}, u_{2}, v_{2} \in V$.

• Complex scalar multiplication on $V_{\mathrm{C}}$ is defined by

  $$(a+b i)(u+i v) = (a u-b v)+i(a v+b u)$$

  for all $a, b \in \mathbb {R}$ and all $u, v \in V$.

Prove that with the definitions of addition and scalar multiplication as above, $V_{\mathrm{C}}$ is a complex vector space.