10  General vector spaces

A vector space needs vector addition and scalar multiplication, but it need not resemble Euclidean coordinate space.

You may be very familiar with Euclidean space \(\mathbb{R}^n\) as a vector space, and quite used to the notion that we have an inner product and a corresponding distance measure on this space. However, linear spaces are much more general. An inner product or norm is not part of the definition of a general vector space. Only the a “linear structure” matters: Consider for example a general vector space over \(\mathbb{R}\): An abstract set \(V\) with operations such that one can add vectors, \[z = x + y \in V,\] and multiply vectors by constants \(c\in\mathbb{R}\), \[z = c x \in V\] where \(c \in \mathbb{R}\). The full set of axioms is as follows. We focus on vector spaces over the field \(\mathbb{R}\), but the definition can be generalized to vector spaces over any field such as \(\mathbb{C}\).

Definition: Vector space. A vector space over the field \(\mathbb{R}\) is a set \(V\) together with a binary vector addition \(+ : V \times V \to V\) and scalar multiplication \(\cdot : \mathbb{R}\times V \to V\) such that, for all \(x\), \(y\), \(z \in V\) and all \(\alpha,\beta \in \mathbb{R}\), the following axioms are true:

  1. There exists a \(0\in V\) such that \(0 + x = x\) for all \(x\in V\)

  2. \(x + (y + z) = (x + y) + z\)

  3. \(x + y = y + x\)

  4. For every \(x \in V\), there exists \(x' \in V\) such that \(x+x'=0\)

  5. \((\alpha\beta)\cdot x = \alpha\cdot(\beta\cdot x)\)

  6. \(1\cdot x = x\)

  7. \((\alpha + \beta)\cdot x = \alpha\cdot x + \beta\cdot x\)

  8. \(\alpha \cdot (x + y) = \alpha\cdot x + \alpha \cdot y\)

In this exercise, we will train our ability to think abstractly, and also try to get acquainted with general vector spaces.

Example: Somerset Apple Cake. Ingredients:

  • 170 g butter

  • 170 g light brown sugar

  • 3 eggs

  • 1 tablespoon liquid honey

  • 1.5 teaspoons cinnamon

  • 0.5 teaspoon cloves

  • 0.5 teaspoon nutmeg

  • 1 teaspoon baking powder

  • 240 g wheat flour

  • 700 g apples, diced

  • 100 ml apple cider, apple juice, or milk

  • 100 g light sultana raisins (optional)

Instructions:

  • Cream the room-temperature butter with the sugar until light and fluffy.

  • Beat in the eggs and honey.

  • Sift the flour, baking powder, and spices. Add them to the mixture and stir until the batter is smooth and lump-free.

  • Stir in the milk or apple cider (and the raisins, if you choose to use them; they can be soaked in cider for a couple of hours beforehand, if desired).

  • Finally, fold the apple pieces into the batter.

  • Pour the batter into a greased round tin with parchment paper at the bottom, or use a deep, round ovenproof dish (24 cm in diameter).

  • Bake the cake in the middle of the oven at 160°C for 1.5 hours (check with a cake tester to ensure it’s fully baked).

  • Serve warm or cold with a dollop of whipped cream, clotted cream, vanilla ice cream, or whatever you like.

Exercises

Exercise 10.1 (Cooking with vectors 1) We can think of an ingredient list as a vector. For example, suppose that our universe of ingredients consists only of

\[ (\text{butter},\text{sugar},\text{eggs},\text{flour},\text{apples}). \]

After choosing suitable units for each ingredient, an ingredient list can then be represented by a vector

\[ x=(x_1,x_2,x_3,x_4,x_5), \]

where each component gives the amount of one ingredient.

  1. Using grams for butter, sugar, flour, and apples, and number of eggs for eggs, write the corresponding five-component vector for the Somerset Apple Cake recipe.

  2. What would the vector \((100,0,2,300,0)\) mean? What does a zero component mean?

  3. What is the dimension of this vector space? Give a natural set of basis vectors and explain what each basis vector represents.

  4. Suppose that \(x\) and \(y\) are the ingredient vectors for two recipes. What does the vector \(x+y\) represent?

  5. What does the vector \(2x\) represent? What about \(\frac{1}{2}x\)? Does the scalar \(2\) have a physical unit?

  6. Our choice of units is part of the way we represent the recipe. Suppose milk were another component. Explain why we should not represent 1 litre of milk as \(1\) in one recipe and 1000 ml of milk as \(1000\) in another without taking account of the different units.

  7. Typical recipes contain ingredients that are not among our five chosen ingredients. How could we enlarge our vector space to accommodate more recipes? What happens to the dimension of the space?

Exercise 10.2 (Cooking with vectors 2) In Exercise 10.1, we represented recipes using a fixed list of possible ingredients. If you did not do this exercise already, please do so first. In this exercise, we make this construction more general.

Let \(S\) be the set of all possible ingredients. An ingredient list can be represented by a function

\[ f:S\to\mathbb{R}, \]

where \(f(s)\) gives the amount of ingredient \(s\). We assume that \(f(s)=0\) for all but finitely many ingredients. The set \(L\) of all such functions is a vector space, with addition and scalar multiplication defined in the usual way.

  1. Explain how this description generalizes the finite-dimensional ingredient vectors used in Exercise 10.1. If \(S\) contains infinitely many ingredients, what can you say about the dimension of \(L\)?

  2. Mathematically, \(L\) contains vectors with negative components. Can such vectors be interpreted as ordinary ingredient lists? What about the zero vector?

  3. Define the subset

\[ L_+=\{f\in L\mid f(s)\geq 0\text{ for every }s\in S\}. \]

Is \(L_+\) a vector space? Which vector-space properties fail? Is \(L_+\) closed under addition? Is it closed under multiplication by nonnegative scalars?

  1. Even if \(f,g\in L_+\) are ingredient lists for two perfectly good recipes, \(f+g\) need not describe a sensible recipe. What information is missing from an ingredient vector?

  2. We might try to describe the steps of a recipe by operators acting on \(L\). Consider, for example, the instruction “cream the butter and sugar”. Can this operation be represented naturally as a linear map \(T:L\to L\) on our present ingredient space? Explain the challenges one encounters.

  3. What additional information might have to be included in our mathematical description before operations such as mixing, creaming, heating, and baking could be represented?

  4. In quantum mechanics, we similarly choose a mathematical space to represent physical states and operators to represent physical quantities and transformations. Why is the choice of the space important for determining what the mathematical model can describe?

Solutions

Solution for Exercise 10.1

  1. With the ordering

    \[ (\text{butter},\text{sugar},\text{eggs},\text{flour},\text{apples}), \]

    the Somerset Apple Cake recipe is represented by

    \[ x=(170,170,3,240,700). \]

    The components do not all have the same units: the first, second, fourth, and fifth components are measured in grams, while the third is a number of eggs. This is not a problem as long as the meaning and unit of each component are fixed.

  2. The vector

    \[ (100,0,2,300,0) \]

    represents an ingredient list containing 100 g butter, no sugar, 2 eggs, 300 g flour, and no apples.

    A zero component means that the corresponding ingredient is absent.

  3. The vector space has dimension 5. A natural basis is

    \[ e_1=(1,0,0,0,0),\qquad e_2=(0,1,0,0,0),\qquad \ldots,\qquad e_5=(0,0,0,0,1). \]

    Each basis vector represents one unit of one ingredient and zero of all the others. For example, \(e_1\) represents 1 g of butter and \(e_3\) represents 1 egg.

    Any ingredient vector can be written as a linear combination of these basis vectors. For example,

    \[ (170,170,3,240,700)=170e_1+170e_2+3e_3+240e_4+700e_5. \]

  4. Addition is performed component by component. If

    \[ x=(x_1,\ldots,x_5),\qquad y=(y_1,\ldots,y_5), \]

    then

    \[ x+y=(x_1+y_1,\ldots,x_5+y_5). \]

    Thus, \(x+y\) represents putting together the ingredient quantities from the two lists. If both recipes require flour, for example, the flour amounts are added.

    The resulting vector is certainly a valid ingredient vector, although it need not correspond to a sensible cake recipe.

  5. Multiplying by 2 doubles the amount of every ingredient:

    \[ 2x=(2x_1,\ldots,2x_5). \]

    This can be interpreted as making twice as much of the recipe. Similarly,

    \[ \frac12x \]

    corresponds to making half as much.

    The scaling factor is dimensionless. It simply tells us by what numerical factor all ingredient quantities are changed.

  6. The numerical value of a component only has meaning together with the unit chosen for that component. If the milk coordinate is defined to be measured in millilitres, then

    \[ 1\ \mathrm{litre}=1000\ \mathrm{ml} \]

    must be represented by the number 1000.

    If we sometimes used litres and sometimes millilitres without converting them, then the same physical amount could be represented by different vectors. We therefore choose one standard unit for each component.

  7. We can add more components, one for each new ingredient. For example, if cinnamon and milk are added to our list of possible ingredients, the vectors would have seven components instead of five.

    Each additional independent ingredient increases the dimension by one. If we allow \(n\) possible ingredients, the corresponding vector space has dimension \(n\).