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Discover Linear Algebra: A first course in linear algebra

Jeremy Sylvestre

Discovery guide 15.1 Discovery guide
A4US

Suppose you have a collection of mathematical objects. The objects in the collection may satisfy all/some/none of the following rules, depending on the objects. In the rule statements, bold variable letters represent arbitrary objects in the collection, and ordinary variable letters represent arbitrary scalars (i.e. numbers).
List 15.1.1. (A) Addition rules
  1. The objects can be added (two at a time), and the resulting “sum” is always equal to another in the collection of objects.
  2. Every \(\uvec{v},\uvec{w}\) satisfy
    \begin{equation*} \uvec{w}+\uvec{v} = \uvec{v}+\uvec{w}\text{.} \end{equation*}
  3. Every \(\uvec{u},\uvec{v},\uvec{w}\) satisfy
    \begin{equation*} \uvec{u}+(\uvec{v}+\uvec{w}) = (\uvec{u}+\uvec{v})+\uvec{w}\text{.} \end{equation*}
  4. There is a special zero object in the collection, so that every \(\uvec{v}\) satisfies \(\uvec{v}+\zerovec=\uvec{v}\text{.}\)
  5. Every \(\uvec{v}\) has an opposite \(\widetilde{\uvec{v}}\) so that \(\uvec{v}+\widetilde{\uvec{v}}=\zerovec\text{.}\)
List 15.1.2. (S) Scalar multiplication rules
  1. The objects can be scaled by a numerical factor (called a scalar), and the resulting “scaled object” is always equal to another in the collection of objects.
  2. Every \(k,\uvec{v},\uvec{w}\) satisfy
    \begin{equation*} k(\uvec{v}+\uvec{w}) = k\uvec{v}+k\uvec{w}\text{.} \end{equation*}
  3. Every \(k,m,\uvec{v}\) satisfy
    \begin{equation*} (k+m)\uvec{v} = k\uvec{v}+m\uvec{v}\text{.} \end{equation*}
  4. Every \(k,m,\uvec{v}\) satisfy
    \begin{equation*} k(m\uvec{v}) = (km)\uvec{v}\text{.} \end{equation*}
  5. Every \(\uvec{v}\) satisfies \(1\uvec{v} = \uvec{v}\text{.}\)

Discovery 15.1.

Read and briefly discuss the rules in your group. In particular, make sure everyone in your group understands the differences between the LHS and RHS in each of Rule A 2, Rule A 3, Rule S 2, Rule S 3, and Rule S 5.
It may help to come up with expressions for these algebra rules in plain English rather than letters and variables. For example, Rule A 2 states that order doesn’t matter in adding objects.

Discovery 15.2.

These rules are modelled on the properties of vectors in \(\R^n\text{.}\) Convince yourself that all the rules are true when the collection of mathematical objects considered is “all vectors in \(\R^2\text{.}\)” In particular, make sure you know what the zero object is in the collection (Rule A 4), and how to determine an object’s opposite (Rule A 5).

Discovery 15.3.

For each of the following collections of objects, convince yourself that all the rules are true. In particular, make sure you know what the zero object is in the collection (Rule A 4), and how to determine an object’s opposite (Rule A 5).

(a)

All \(2\times 2\) matrices.

(b)

All \(m\times n\) matrices. (Here \(m\) and \(n\) are some specific but unknown numbers.)

(c)

All polynomials in the variable \(x\text{.}\)

(d)

All polynomials in the variable \(x\) of degree \(2\) or less (i.e. no \(x^3\) or higher allowed).

(e)

All real numbers.

Discovery 15.4.

Suppose you have a collection of objects that satisfies all of the rules. (Don’t pick a specific example collection, just think in the abstract.)

(a)

For an object \(\uvec{v}\text{,}\) is it necessarily always true that \(\zerovec + \uvec{v} = \uvec{v}\text{?}\)
Hint.
Look at Rule A 2 and Rule A 4.

(b)

For an object \(\uvec{v}\) and its opposite \(\widetilde{\uvec{v}}\text{,}\) is it necessarily always true that \(\widetilde{\uvec{v}} + \uvec{v} = \zerovec\text{?}\)
Hint.
Look at Rule A 2 and Rule A 5.

(c)

By Rule A 5, every object has an opposite which itself is an object. What is the opposite of an opposite? Make sure you can justify that your answer satisfies the definition of opposite contained in Rule A 5.

(d)

Suppose \(\uvec{v}\) is an object. What object do you think \(0\uvec{v}\) should be equal to? Do the rules provide direct evidence to support your guess?

(e)

Here is a justification of your guess from Task d. (Assuming you guessed correctly!) Fill in the blanks with the identifier of the rule that justifies each step, working down the left-hand side first. Make sure you understand how and for what objects that rule is being applied.
\begin{align*} \uvec{v}+\widetilde{\uvec{v}} \amp= \zerovec \amp \amp\fillinmath{XXXXX} \amp 0\uvec{v}+(1\uvec{v}+\widetilde{\uvec{v}}) \amp= \zerovec \amp \amp\fillinmath{XXXXX}\\ 1\uvec{v}+\widetilde{\uvec{v}} \amp= \zerovec \amp \amp\fillinmath{XXXXX} \amp 0\uvec{v}+(\uvec{v}+\widetilde{\uvec{v}}) \amp= \zerovec \amp \amp\fillinmath{XXXXX}\\ (0+1)\uvec{v}+\widetilde{\uvec{v}} \amp= \zerovec \amp \amp\text{(arithmetic)} \amp 0\uvec{v}+\zerovec \amp= \zerovec \amp \amp\fillinmath{XXXXX}\\ (0\uvec{v}+1\uvec{v})+\widetilde{\uvec{v}} \amp= \zerovec \amp \amp\fillinmath{XXXXX} \amp 0\uvec{v} \amp= \zerovec \amp \amp\fillinmath{XXXXX} \end{align*}

(f)

Use the rules to “simplify” the expression \(\uvec{v} + (-1)\uvec{v}\text{.}\) Make sure each step is justified by a specific rule, similarly to Task e.
Hint.
Start by using Rule S 5 backwards, as used to transform the first line to the second in Task e.

(g)

Take \(\uvec{v} + (-1)\uvec{v} = X\text{,}\) where \(X\) is your final simplified expression from Task f. We can “cancel” the \(\uvec{v}\) from the LHS by adding \(\widetilde{\uvec{v}}\) to both sides of the equality. Based on the resulting equality after doing that, what do you think is a better name for \(\widetilde{\uvec{v}}\) than opposite of \(\uvec{v}\)?

Discovery 15.5.

Nominate one member of your group to become an object, and consider the collection of objects that consists of just one object (namely, the group member you nominated).

(a)

Can you come up with some sort of addition so that Rule A 1 is true?

(b)

Can you come up with some sort of scaling operation so that Rule S 1 is true?

(c)

Check whether the other eight rules hold true with the operations you have devised in this activity.
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