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Section 2.1 Additive Inverses

Definition 2.1.1.

A linear space is a set \(\mathbb{L}\) containing objects called points where for any points \(\mathbf{P}\) and \(\mathbf{Q}\) in \(\mathbb{L}\) and any real number \(t\) there exist unique points:

\begin{align*} \amp \bullet \amp \amp\mathbf{P}+\mathbf{Q} \amp \amp \text{called the sum of } \mathbf{P} \text{ and } \mathbf{Q} \\ \amp \bullet \amp \amp t \mathbf{P} \amp \amp \text{called the scalar product of } t \text{ and } \mathbf{P} \end{align*}

and where the following axioms are satisfied for any points \(\mathbf{P}\text{,}\) \(\mathbf{Q}\text{,}\) \(\mathbf{R}\) \(\in \mathbb{L}\) and any \(a, b \in \mathbb{R}\text{:}\)

  1. \(\displaystyle \mathbf{P}+\mathbf{Q}=\mathbf{Q}+\mathbf{P}\)

  2. \(\displaystyle \mathbf{P}+(\mathbf{Q}+\mathbf{R})=(\mathbf{P} + \mathbf{Q})+\mathbf{R}\)

  3. \(\displaystyle a(b\mathbf{P}) = (ab)\mathbf{P}\)

  4. \(\displaystyle (a+b)\mathbf{P} = a\mathbf{P} + b\mathbf{P}\)

  5. \(\displaystyle a(\mathbf{P}+\mathbf{Q}) = a\mathbf{P} + a\mathbf{Q}\)

  6. There exists a point \(\mathbf{0}\text{,}\) called the additive identity, such that \(\mathbf{P}+\mathbf{0}=\mathbf{P}\)

  7. \(a\mathbf{P}=\mathbf{0}\) if and only if \(a=0\) or \(\mathbf{P}=\mathbf{0}\)

From this definition, the goal is to discover as many consequences as possible. This often involves working with equations. When working with equations one side of an equation may be changed if the other is changed in exactly the same way. Also, an object may be substituted with another to which it is equal.

For points \(\mathbf{P}\text{,}\) \(\mathbf{Q}\text{,}\) and \(\mathbf{R}\) in linear space \(\mathbb{L}\text{,}\) which of the following does the definition of Linear Space Definition 2.1.1 guarantee are also points in \(\mathbb{L}\text{?}\)

  1. \(\displaystyle 1\mathbf{Q}\)

  2. \(\displaystyle 1+\mathbf{Q}\)

  3. \(\displaystyle \mathbf{P}\mathbf{Q}\)

  4. \(\displaystyle 3\mathbf{P} + -1\mathbf{Q}\)

  5. \(\displaystyle \mathbf{P} - \mathbf{Q}\)

  6. \(\displaystyle 3\mathbf{P} + 5\)

  7. \(\displaystyle \mathbf{P} + \mathbf{Q}+ \mathbf{R}\)

Does \(\mathbf{0} = 0\) ? Explain.

For \(\mathbf{Q}\) in linear space \(\mathbb{L}\text{,}\) and \(c \in \mathbb{R}\text{,}\) does it follow that \(0\mathbf{Q}=\mathbf{0}\text{?}\) Does it follow that \(c\mathbf{0} = \mathbf{0}\text{?}\) Is there a property that says \(1\mathbf{Q} = \mathbf{Q}\text{?}\)

Definition 2.1.6.

Let \(\mathbf{Q} \in \mathbb{L}\text{.}\) The additive inverse of the point \(\mathbf{Q}\) is the point, written \(-\mathbf{Q}\text{,}\) that when added to \(\mathbf{Q}\) results in the additive identity element, \(\mathbf{0}\in \mathbb{L}\text{.}\) In symbols: \(-\mathbf{Q}\) is the point such that \(\mathbf{Q} + -\mathbf{Q} = \mathbf{0}\text{.}\)

The symbol "\(-\)" is now used to indicate the additive inverse of a real number and the additive inverse of a point which makes \(-1\mathbf{Q}\) ambiguous. Describe what is mean by \((-1)\mathbf{Q} \) and how it differs from \(-(1\mathbf{Q}) \text{.}\)