Quantity – Different Kinds of Numbers: Vectors – it really all started with this post.  But I didn’t know it yet at the time…

Groupings, Shopping Lists, and Vectors – part 1 – where we take off on a journey: why vector algebra might be important, and how come the traditional textbook approach sucks most of the juice out of it.

Groupings, Shopping Lists, and Vectors – part 2 – a shopping scenario as a starting point for what is essential about vectors: that you know which stuff and how much of each.

Groupings, Shopping Lists, and Vectors – part 3 – a natural notation for vectors different from that in textbooks, which we might call annotated vectors

Groupings, Shopping Lists, and Vectors – part 4 – vector addition seen as combining shopping lists or cash register receipts

Groupings, Shopping Lists, and Vectors – part 5 – vector addition as an entry point into the distributive property

Groupings, Shopping Lists, and Vectors – part 6 – extended prices: introducing inner products

Groupings, Shopping Lists, and Vectors – part 7 – various situations where inner products show up

Groupings, Shopping Lists, and Vectors – part 8 – weighted averages of test scores: matrix times a vector

Groupings, Shopping Lists, and Vectors – part 9 – matrix times a vector, shown in 3D!

Groupings, Shopping Lists, and Vectors – part 10 – matrix multiplication, starting from a shopping list

Groupings, Shopping Lists, and Vectors – part 11 – a further look at matrix multiplication

Groupings, Shopping Lists, and Vectors – part 12 – projective drawings described in terms of matrix multiplication

Groupings, Shopping Lists, and Vectors – part 13 – revisiting vectors and matrices: how do they relate?

Groupings, Shopping Lists, and Vectors – part 14 – how does this treatment compare with textbook treatments?

Groupings, Shopping Lists, and Vectors – part 15 – a look at Excel, showing extended price calculations

Groupings, Shopping Lists, and Vectors – part 16 – vector inner products, shown in Excel

Groupings, Shopping Lists, and Vectors – part 17 – showing matrix multiplication in a regular pattern using Excel

Groupings, Shopping Lists, and Vectors – part 18 – why matrices in textbooks are square.  first look at solving equations

Groupings, Shopping Lists, and Vectors – part 19 – more on solving equations and square matrices

Groupings, Shopping Lists, and Vectors – part 20 – introduction to transformations, using pounds/shillings/guineas

Groupings, Shopping Lists, and Vectors – part 21 – graphing of pound/shilling/guinea situations – showing coordinate axes that are not at right angles

Groupings, Shopping Lists, and Vectors – part 22 – more graphing of pound/shilling/guinea situations

Groupings, Shopping Lists, and Vectors – part 23 – looks at linear combinations, and a geometric model for vector addition

Groupings, Shopping Lists, and Vectors – part 24 – looks at money exchange as a vector addition

Later in the series, we looked at simple everyday scenarios like getting the total cost for the items on the shopping list, which comes from an encounter of the shopping list with a price list; the name for that encounter is inner product.  There are many other scenarios in which inner products show up, and it led us to matrix multiplication, which we looked at from different angles.  More recently, we examined how come almost all matrices you encounter in school are square, meaning they have just as many rows as they have columns.  Since part 20, we’ve been playing with graphical representations involving pounds, shillings and guineas.  Mostly, we played with the fact that one guinea has the same value as one pound and one shilling together.  This fact alone let us to transformation matrices and coordinate systems where the axes are at angles other than right angles.  In part 23 we pictured vectors as arrows, each with a length and a direction (but without a fixed beginning point) and then saw how a particular value – like 2 guineas and 1 shilling – could be pictured as a series of 2 guinea moves and 1 shilling move, performed in any order.

These moves get their meaning from a coordinate system, in this case, with pounds on the horizontal axis and shillings on the vertical access.  The blue arrow represents “one more guinea”, the green arrow represents “one more shilling”, whereas “one more pound” (not pictured) would be represented by an arrow pointing to the right.

In order to get to this point, I’ve had to state several times that what we’re dealing with here is an oversimplification of the English system of currency.  Not only have I left the penny entirely out of the picture, but worse, I’ve been completely side-stepping the fact that one pound is worth 20 shillings.  If this has been hard for you to take, and if you’ve been yelling at the screen “why doesn’t he just convert everything to shillings (or to pounds, or to pence)”, I can sympathize.

We are now in the perfect place to deal with pounds being twenty shillings.  We will introduce a new arrow, a new vector, on this same coordinate grid, and this new vector will represent an exchange of 1 pound for 20 shillings.

In the graph above, the red arrow represents the basic exchange of a single pound into twenty shillings.  As a result of applying this exchange move, you end up with one pound less, and twenty shillings more.   If we look at actual collections of pound and shilling coins, the pound (black) arrow, the shilling (green) arrow, the exchange (red) arrow, show what happens when you add a pound, add a shilling or exchange a pound for twenty shillings.  The same arrows, but pointing in the opposite direction, would represent what happens if you take away one pound, take away one shilling, or exchange twenty shillings for one pound.  All of these are pretty straightforward.  Also shown is the “guinea move”, but in terms of actual collections of pound and shilling coins, the guinea move is somewhat ambiguous.  Since the guinea only represents a value, and not a particular combination of bills or coins, identifying the guinea with one pound and one shilling is a bit arbitrary.  We could have pictured the guinea as an arrow going straight up for 21 shillings.  Doing it the way we did here is not much of a limitation, though: the guinea move, as shown, followed by the exchange move, will have the net effect of adding 21 shillings to whatever the starting point was.

When we think of actual collections of pound and shilling coins, we know that an exchange of a pound for twenty shilling isn’t entirely without consequences.  Though the exchange may be free, it may require you to go to a bank or a store, or some person on the street, and ask them to make the exchange for you.  For this, the bank needs to be open, and the store needs to be open and willing, and the person on the street needs to be willing and happen to have the right number of coins on hand.  If there are machines that make the exchange for you, without charging a fee, then still somebody has to feed those machines and maintain them.  The true free unhampered exchange of coins for other coins only exists as an abstraction, a simplification, an illusion.  But it is useful simplification.  It is useful to say that one pound equals twenty shillings.  or we could say, more precisely, that one pound exchanges for twenty shillings, and that such an exchange is often easy and cheap, and that we can often ignore the cost (i.e. the hassle) of doing so.

We could say things like:   , where indicates a pound move, indicates a shilling move, and indicates an exchange move of one pound for 20 shillings.  Each represents a different path, starting from the origin, and all ending up at the 87 shilling mark.  All represent the same value, the value in Jane’s jar, assuming that the exchanges are free and unlimited.

The graph above, from part 21, shows axes of guineas and shillings.  The axes are not at right angles, but doing so wasn’t totally arbitrary: if you move one unit horizontally to the right, you add one pound to the value (a guinea is one pound plus one shilling).  To find the value of Jane’s jar in terms of guineas and shillings, you would follow the grid lines, and see that it amounts to 4 guineas and 3 shillings.

We could interpret this same thing in terms of movement.  If we start at the place where the axes meet (usually referred to as the origin), and we make 4 ‘guinea’ moves, and 3 ‘shilling’ moves, we arrive at the point representing Jane’s jar.  Each of the 4 guinea moves is a move in a particular direction (here, diagonally up to the right) and of a particular distance.  Each of the 3 shilling moves is a move in a particular direction (here, straight up) and of  a particular distance.

From our starting point at the origin (meeting point of the axes), any combination of 4 guinea moves and 3 shilling moves will get us to the same ending point.  We can take 1 guinea move, 1 shilling move, 3 guinea moves and finally 2 shilling moves, and end up at Jane’s jar.  Or we could take 2 guinea moves, 1 shilling move, 1 guinea move, 2 shilling moves, and finally 1 guinea moves, and also end up at Jane’s jar.  All that matters is how many guinea moves we did altogether, and how many shilling moves we did altogether.   This can be expressed in shorthand, using a notation I haven’t used here before.  Let’s write for a single guinea move, and for a single shilling move.  Then we can write:

(a) Jane’s jar =

(b) Jane’s jar  =

(c) Jane’s jar =

(d) Jane’s jar =

(e) Jane’s jar =

and more.

In the old tradition of pirate treasure maps, “go 100 paces North East,” each of the expressions in that list represents a particular path, though all happen to end up in the same place.

The above represents an informal introduction to the geometric notions of vectors, vector addition, and multiplication of vectors by a number.   Textbooks often introduces vectors as arrows with direction and length, like our ‘guinea move’ or like our ‘shilling move’.  They will tell you that to add two vectors, you need to draw the parallelogram with the two vectors as their sides, and then draw the diagonal of the parallelogram, and this diagonal represents the sum.  They really are just talking about the result of two motions one after another.

In the diagram above, the blue arrow represents , the green one represents .  In the box on the right, three different ways to combine 2 guinea moves and one shilling move are shown.   Each path, if started at the origin, will end up in the same place.  That place can be characterized as .  When you focus on the geometry, and divorce yourself from the specific meaning of guinea values and shilling values, you can see that what we have shown here matches the textbook treatment of a vector as an arrow with direction and length (but with flexible starting point), and matches the textbook treatment of vector addition as the joining of two arrows end-to-start.  Moreover, the idea generalizes to taking any number of the first vector, and any number of the second vector, and joining them all together, in what is known as a linear combination of the two original vectors.

We explored how the vector representation of a value in one system can be obtained from that in the other system through matrix multiplication, where the matrix used (the transformation matrix) is a square matrix (that is, has the same number of rows as it has columns).

In the last post, we used graphs to represent values.  In the first graph, there is a pound axis and a shilling axis, drawn at right angles.  Values are represented by a point.  In the second graph, the shilling axis is left the same, the values are all drawn in the same place as before, but values are now expressed in terms of guineas and shillings.  This can be done, at the expense of showing the guinea axis not at right angles but at a different angle to the shilling axis.  We saw that, though axes at right angles are convenient, there is nevertheless some arbitrariness about it.

In this post, we will pursue that by drawing graphs again, but this time with the guinea axis at right angles to the shilling axis.

On the guinea and shilling axes, the value of Jane’s jar is represented as shown, indicating 4 guineas and 3 shillings.  As you can see, this graph can not be overlaid on top of the ones from the previous post: Jane’s jar is in a different position now.  The value in Jane’s jar is still 7 shillings more than 4 pounds; this is shown in the graph by having the 4-pound mark 7 shillings below the mark for Jane’s jar.   This is consistent with the notion that a pound is one guinea minus one shilling, and showing pounds below the guineas axis is a way to express their values as having a negative shilling component.   We can show the same values in the same place, using shilling and pound axes, but only if we are willing to accept pounds and shilling axes to be at something other than right angles; this is shown below.

This graph, when printed on transparent paper, could be overlaid over the previous graph, and show both the shilling axis and the value in Jane’s jar in the same location.   As you see, ‘guineas’ are no longer marked on it; we merely have pounds and shilling axes, just not at right angles.  Even though some of the four graphs shown (two in this post, two in the prior post) look more familiar than others, you could argue that none is intrinsically ‘truer’ than the others.

The coordinate grids where axes are at non-right angles will give us an interesting lead in to vector addition and linear combinations, viewed geometrically.

Our starting point in that post was that values expressed in pounds and shillings can also be expressed in guineas and shillings, and vice versa.  There is a way to show this graphically, and doing so gives an opportunity to show coordinate systems that are strange to many people: coordinate systems where the axes are not at right angles to one another.

In the graph, we display pounds on the horizontal axis, and shillings on the vertical axis.  The value of bills and coins in Jane’s jar is indicated in the graph, and you can read that value as 4 pounds and 7 shillings.  If you are used to writing coordinates as pairs like this: (4,7), you can do that – or you can write it as a vector like this:

On the graph we also indicated points for 1 guinea, 2 guineas, etc., and we can think of the line on which these lie as a guinea-axis (neither horizontal nor vertical).  The value in Jane’s jar can be expressed in guineas and shillings, and this can be done by noticing that the value in Jane’s jar is precisely 3 shillings above the marker for 4 guineas.  Written as a vector, this would be

and it is completely valid to think of this as a coordinate pair based on the guinea-axis and the shilling-axis.  Though the guinea-axis and the shilling-axis are not at right angles, it doesn’t take long to get used to those kinds of axes, and read values accurately in terms of guineas and shillings straight from the graph.  What I think is genuinely confusing, and not just a matter of familiarity, is to write those coordinates as (4,3).  This is confusing, since I might now think of that as 4 pounds and 3 shillings, which conventionally would also be written as (4,3).  But that confusion arises from the (4,3) notational system, and would be absent if we were used to writing (4 guineas, 3 shillings) instead.

It is useful to separate out these two things: (1) notational confusion, and (2) conceptual confusion.  Notation confusion comes from common notations, like (4,3) which assume agreement on what the components mean.  Conceptual confusion might arise from being unfamiliar with axes that aren’t at right angles with one another.  In the graph below, the “pound” axis is no longer explicitly shown, and values can be interpreted as having two coordinates according to the two coordinate axes shown: the guinea axis and the shilling axis.

Compared to the earlier graph, the value of money in Jane’s jar shows up in the exact same place.  If the graphs were drawn on transparent paper, and held directly one above the other, you would see the origin, the shilling axis, the guinea axis, and the Jane’s jar point all line up.  The graph is a bit harder to read, but mostly because we aren’t as familiar with this kind of graph as we are with those where axes are at right angles.  To read the coordinates corresponding to Jane’s jar, we would follow the grid lines (shown parallel to the axes) and see where they intersect the axes.   The vertical grid line through Jane’s jar hits the guinea axis at the 4 guinea mark, and the not-horizontal grid line through Jane’s jar hits the shilling axis at the 3 shilling mark.

Our preference for axes at right angles isn’t entirely arbitrary, but can still be seen as an inherited convention.  When it comes to which axis is labeled the “x” axis and having positive values on the right and negative values on the left – most of us are clear that this is a convention, a cultural heritage, something you and I do simply because everybody else does it the same way.  It is important to know and to honor those conventions, because deviating from these conventions risks having your graph be misunderstood by just about everybody else.  Yet at the same time it is useful to be clear that there is no deep mathematical reason that requires “x” to point towards the right, or axes to be at right angles.  There are consequences, though, for having axes at right angles, as there are consequences for having the scales on x and y axes be the same.  If you are familiar with the standard equation for a circle (centered at the origin) as , where indicates the radius, you will likely also be aware that the circle will degenerate into an ellipse if the scale on the “x” axis doesn’t match the scale on the “y” axis.   Additional distortions of shape will show up if you graph familiar equations (e.g. ) on axes like those on our guinea/shilling graph and expect to see the shapes familiar from school.

Yet when we started out graphing pounds and shillings, our decision to show those at right angles is essentially arbitrary.  Once we made that choice, the guinea axis showed up at a funny angle.  There is nothing intrinsically “right-angled” about pounds and shillings.  We could have started with values expressed in guineas and shillings and we could have chosen to represent those geometrically in a “normal-looking” coordinate grid based on guineas and shillings at right angles.  We will follow this idea in the next post.

There is a term in use that is of a different nature: the guinea.  The guinea is not a name for a coin or a bill, but the name for a value.  One guinea represents the same as one pound and one shilling, or 21 shillings.  A Wikipedia article says that guineas are used by art dealers and lawyers to quote their prices, and British horse races still express their prize funds in guineas.  Guineas can also be used if a value is not a whole number of guineas.  You can have a fifth of a pound (4 shillings), and you can likewise have a third of a guinea (7 shillings).   The convenience of dividing by three is sometimes offered as a reason why the guinea is still around.  For somebody raised outside of this culture, like me, it isn’t easy to appreciate the appeal of having both pounds and guineas around (even if the guinea doesn’t exist as a coin or bill).

The relationship between pounds and shillings and guineas gives us an interesting background in which to talk about expressing things in terms of different units.  The process of stating something in new units is often called transformation, and I’ve suggested earlier that transformations are a frequent source for encountering matrices, and specifically square matrices.

I’m going to take the shilling/pound/guinea situation as a starting point.  I’m not suggesting that any of the things to follow here will be things that Englishfolk are actually discussing doing.  For one, I’m going to ignore the penny altogether.  To be more precise, my starting point will be that we have three units of value, and one of these is worth the sum of the other two.   I’m even going to ignore that we know the pound to be worth 20 shillings.  (The things I’m ignoring could be brought back into the picture later, without too much difficulty.  For now, I’m interested in keeping the example and the numbers simple.)

In this hypothetical England, I could represent values as a vector in different ways.  Let’s start with some number of guineas, and some number of shillings.  Below is an example:

Here, the amount is 3 guineas and 5 shillings.  This same value could be re-stated in terms of pounds and shillings.  This is important: the same amount, but in a different representation.

It is not particularly difficult to convert from one representation to the other, in either direction.  The conversion is simple enough that any mention of matrix multiplication would sound like absolute overkill.  Still, this example is useful to introduce the structure of a transformation matrix, and this will help us when we look at more practical examples where the conversion isn’t quite so trivial.

Let’s first look at conversion to the pounds/shilling representation.  Shown above is each of the units of the guinea/shilling representation, expressed in the pounds/shilling representation.  A single guinea is one pound and one shilling; and a single shilling is zero pounds and one shilling.  These two vectors can be combined in a matrix:

and this matrix is called a transformation matrix.  The conversion from guinea/shilling to pound/shilling can now be expressed as a matrix-vector multiplication using this transformation matrix:

where the original representation is on the top right, the transformation matrix is on the left, and the resulting representation in pound/shilling is on the right.  You can check to see that it works, and that it will work for any amounts expressed in guineas and shillings.  Yes, matrix multiplication is overkill here, but it does work and it does allow us to illustrate something useful.

To finish this post, let me briefly show how conversion in the opposite direction would work – conversion from a value in the pound/shilling representation to a value in the guinea/shilling representation.  (Note: for the moment I’m ignoring that a pound equals 20 shilling altogether.)  The conversion from pound/shilling to guinea/shilling can also be expressed using a transformation matrix, and we will look at the relationship between the two transformation matrices in the next post.  In this post, I’ll merely introduce it:

The transformation matrix reflects (in its pound column) that one pound equals one guinea minus one shilling, and reflects (in its shilling column) that one shilling equals zero guineas plus one shilling.  You can verify that the matrix multiplication works, and will work for other pound/shilling amounts (as long as you allow negative shilling amounts and continue to ignore the issue of breaking pounds into shillings).

Let’s use our Bill Amend comic strip once again, and look at the problem as stated by the brother in the third panel.

There is a cost per shirt and a cost per sweater, and we don’t know these costs.  They are the things we’re trying to find out.  What we do know is the cost of two shirts and a sweater – that is $60.  Similarly, we know that the cost of one shirt and two sweaters is $75.  If we treat this exactly the same way we’ve been dealing with fast food orders in prior posts, e.g.  in part 6, we could show this as follows:

We’re looking for the numbers in the blue vector that will have the green vector come out just right. This means that both the total price for the first order and the total price for the second order, priced out from the order amounts and the (as yet) unknown prices, must come out to $60 and $75 respectively.  In math class, we’d write:

and assume that somehow we will keep track of what is what.   The things you learned to do in math class, like doubling the second row and then subtracting the first row from the second, these all have counter parts in terms of shirts and sweaters and orders.  Doubling the second row amounts to doubling the second order: 2 shirts plus 4 sweaters will cost $150.  From the first order, we know that 2 shirts plus 1 sweater costs $60.  In double the second order, we have the same number of shirts, but three more sweaters.  For those three extra sweaters, we ended up paying $150-$60, which amounts to $90.  If these three sweaters cost $90, then a single sweater will cost one-third of that, $90/3, which is $30.  So we found the price of a single sweater, which is $30, and we can use either the first order or the second order to recover the price of a single shirt.  If we use the first order, we’d see that two shirts plus the $30-dollar sweater cost us $60, so we must have paid $30 for those two shirts, or $15 per shirt.  (Once we knew that a sweater cost $30, we could also have used the second order to figure out how much one shirt cost: one shirt plus double the 30 dollars amounts to $75, so the shirt part of that order must have been $75 – $60, or $15.)

The steps involved in solving these equations, at least till the point where we nailed one of the unknown numbers, all correspond to row operations on the matrix and known vector.  In a system of equations, written in either of the ways I’ve shown, you can freely multiply a row by any number you like, and freely replace a row by the sum or difference of that row and another row.

Yet you may have solved the system of equations by thinking about the shirts and the sweaters differently.  Is it easy to establish which costs more, a shirt or a sweater? I think it is, since I can imagine walking the first order to the cash register, seeing that it costs $60, and then putting a shirt back on the shelf and grabbing an extra sweater instead.  The order now costs $75, and the extra $15 must come from the extra sweater costing $15 more than the shirt I put back on the shelf.  So a sweater costs $15 more than a shirt.

This still doesn’t exhaust the ways in which you might have reasoned about shirt prices and sweater prices.  You may have noticed that if you first walk the first order to the cash register and then the second, you end up with a total of 3 shirts and 3 sweaters, for which you would have paid a total of $135.  From that, you might conclude that a single shirt and a single sweater must cost $135/3 or $45.  The first order is like a single shirt and a single sweater – with an extra shirt.  The difference between $60 and $45 must account for the single shirt.  Similarly, you might have noticed that the second order is like a single shirt and a single sweater, but with an extra sweater.  This extra sweater cost $75-$45 = $30, so sweaters cost $30.

How did Paige think about the shirts and the sweaters?  The comic strip doesn’t really tell us one way or the other.  What the comic strip does suggest very strongly is that Paige can think in terms of shirt and sweaters, but not in terms of x and y.

For us, there is a similar issue whenever we see a matrix.  What does each row stand for?  What does each column stand for?  If we don’t know – if we can’t talk about the second row and the first column of the coefficient matrix as the number of shirts in the second order – then we can’t do better than talk about the number in row 2 and column 1.  Just maybe, the amazing thing is that some people do not get confused.

If you think back to the order matrix in the part 16, you can see that for it to be square, the number of orders has to somehow be exactly the same as the number of items on the menu.  That situation would be a total coincidence, and would last only until the next car shows up at the order window.  It appears then that the matrices in textbooks must arise from entirely different scenarios than the ones we’ve played with till now.

What are the scenarios that underlie the matrices found in textbooks?  It turns out that this is not a trivial question, or at least it doesn’t have a trivial answer.  For the textbooks often don’t tell you where a matrix comes from.   They may not care, or know, particularly.

My impression is that most matrices in textbooks come from two broad application areas.  One, systems of equations, and two, transformations.  For those broad application areas, we can justify why many – if not most – of the matrices involved would be square.

Let’s take a quick look at equations (and leave transformations for another day).  An equation like 2x+y=60 has many solutions.  There are lots of combinations of numbers so that double the first, added to the second, gives us 60.  For example, 1 for x and 58 for y will do the trick.  But so does  10 for x and 40 for y.  If I think of the equation as a clue for what x and y must be, the clue isn’t powerful enough to nail down x and y.  But if I have another clue, another hint, like x+2y=75, together these hints may be enough to nail down x and y precisely.  Together, these clues give us what is called a system of equations, and they would normally be written in math class like this:

By the time students get to matrix algebra in school, this same system of equations would now be written like this:

where the group of 2 by 2 numbers on the left is called the matrix of coefficients, the vector with the x and y in it is called the vector of unknowns, and the vector on the right is called the known vector, or the right hand side vector.  This use of  a matrix and vectors is consistent with the notion of matrix multiplication, but at this point the matrix and vectors are often introduced simply as a shift in the notation for the system of equations.  it is shorter, more compact, especially if you go through the steps of what is known as Gaussian elimination.  If the system of equations can be solved, it turns out that the solution depends on the known vector in an interesting way.  This way can itself be expressed in matrix notation, using what is called an inverse matrix, and we can write

I’m skipping a bunch of steps on purpose here, including non-trivial ones like how we would find this inverse matrix in the first place.  Here, my main interest is in recovering (imputing) the logic of the progression of topics and techniques in traditional textbooks of matrix algebra.

If square matrices come from systems of equations, where do the systems of equations come from?  In many textbooks, systems of equations are simply the starting point – they appear as if dropped from the sky.  Let’s take another look at the following comic strip (we encountered it earlier in this post)

The brother in the strip can at least come up with a half-way reasonable scenario that might have given rise to the system of equations shown above.  In doing so, he is at a disadvantage: he has to make something up.  He’s working backwards.  He is making up a “story problem” working backwards from the system of equations.  You can’t blame him for coming up with something that – even though Paige can relate to it  – is still kind of lame.  In what real-life situation would you really know (and remember) the cost of two shirts and a sweater,  as well as the cost of one shirt and two sweaters – but not remember the price of each item?   If the price of a sweater and the price of a shirt are not known to us, and can be recovered only by solving a system of equations, it is only because they have been deliberately hidden from us – and what store would have any reason to do that?   But many people like puzzles, and we could view this as a puzzle, the type of puzzle we would call a math puzzle.

From the systems of equations and the matrices in math class, you might never ever guess that vast amounts of money and vast amounts of computer resources are used every day all across the world to perform matrix operations – operations on square matrices even, operations above and beyond the matrix-vector inner product stuff like pricing out orders in the way we saw in the prior parts of this series.

Let me end by sketching a somewhat more realistic example of a problem where you end up with a system of equations.  You test a sample of concrete that is supposed to have a certain amount of steel in it.  Concrete is cheap, and steel is expensive but crucial to the strength of the concrete.  Your supplier would have had incentives to skimp on the amount of steel used.  You know how much pure steel weighs per cubic inch, you know how much pure concrete weights per cubic inch, and you have measured the weight and the volume (cubic inches) of your sample.  What is the composition of your sample?

The structure of the concrete/steel problem isn’t all that different from the shirt/sweater problem.  It is different in that for the shirt/sweater problem you might simply call the store, or look them up on the web.  In the concrete/steel problem you might go ahead and destroy the sample to look at the steel inside, and this might well be a good thing to do.  Yet solving the system of equations would be a quick way to establish that there is insufficient steel in the concrete.

Testing the purity of drugs (whether the legal or illegal kind) can be done by similar techniques.

Do you have better examples of solving systems of equations?  The criteria I’m looking for are (1) that it’s real-life, and (2) easy to state for a non-specialist audience.

Above you see one approach, though not one I recommend.  First, let’s see what we’ve got here.  The blue section represents the menu, containing price and nutritional information for a series of items.  The yellow section represents orders, the green section represents totals.  As before, I ignore taxes in the total price.  (Or if you prefer, the price shown is the price before sales tax).  Each section is what is called a matrix: it has rows and columns, and the meaning of each row as well as the meaning of each column is clear.

The blue matrix has 8 rows of numbers, and three columns of numbers; the yellow matrix has 8 rows of numbers and 2 columns of numbers; and the green matrix has three rows of numbers, and two columns of numbers.

What works about the spreadsheet above is that columns G and H work exactly the same way.  In fact, I can copy the whole column H and then right-click on column I and select “Insert Copied Cells” and I will get another column, with order amounts in yellow, and totals in green, and if I change the label from “Jerry’s order” to “Jane’s order”, and change the amounts from Jerry’s amounts to Jane’s amounts, then the totals for Jane will adjust themselves accordingly.  You can see how this works by looking at the formula in G13.  This formula is shown in the box right above the orange-highlighted G: it shows =SUMPRODUCT($C$4:$C$11, G$4:G$11).   It calculates the inner product of the  price column with the order amount column, giving the total price.  This formula, when copied and pasted into H13, will land there as =SUMPRODUCT($C$4:$C$11, H$4:H$11), which means it will still reference price information, but using Jerry’s order amounts.  So this is what works about the approach shown.

What doesn’t work very well about the approach shown above is that the formulas for G14 and G15 cannot be derived from the one in G13 by copy and paste.  Even though we tried very hard to protect the row numbers by typing $C$4 so that when we copy the formula downward Excel won’t mess up and turn it into C5:C12, we can’t get the column designation right.  Neither $C4 nor $C$4 works to get it to paste as D4.  This is not surprising.  Excel adjusts formulas by noticing how far over and down the target cell is from the source cell.  It has no way of guessing that you wanted the columns to advance as you move the cells down.  So in the spreadsheet shown above, what is typed into G14 is =SUMPRODUCT($D$4:$D$11,G$4:G$11) and in G15 we have =SUMPRODUCT($E$4:$E$11, G$4:G$11).

Below, I show a set up that fits Excel’s way of doing things better, and just maybe this is easier for human beings as well.

If you focus on the totals matrix, in green, you see that it gets its columns from the blue matrix, and its rows from the yellow matrix.  The order matrix (yellow) is now 3 rows by 8 columns of numbers, the blue matrix is 8 rows by 3 columns of numbers, and the totals matrix is 3 rows by 3 columns of numbers.  Yet you can see clearly that the fact that the totals matrix has the same number of rows as columns is mostly coincidence: all it takes is adding or removing a single order, and the totals matrix changes its number of rows accordingly.  Conversely, if we were to add another column of, say, cholesterol data to the blue matrix, it would change the shape of the green matrix accordingly.  The green matrix has the same columns (not just the number of columns) as the blue matrix, and the same rows (not just the number of rows) of the yellow matrix.

I wish I could tell you that the content to be typed into Joe’s total price cell would be =SUMPRODUCT(B13:I13,K4:K11) – but Excel lets us down here.  Though Excel documentation (Office Excel 2007) suggests that SUMPRODUCT works on any two ranges of numbers as long as they are equally long, and though we’ve seen before that Excel will blithely calculate numbers without regard for whether they make any sense or not (we saw in the previous posts that Excel will gladly calculate the inner product of two orders), Excel nevertheless refuses to calculate the inner product of a row with a column using SUMPRODUCT.  That is too bad, and it is a restriction that I would consider to be a bug in the program.  Fortunately, you can see in the figure above that =MMULT(B13:I13,K4:K11) does the trick.  MMULT, which appears to stand for matrix multiplication, not only accepts that one range is a row and the other one is a column, it seems to insist on it.  We’re over the hump, though, and the road is downhill from here.  In thinking through the details of =MMULT(B13:I13,K4:K11) we can see that we want to protect the columns B through I, as they indicate the range of items on the menu: they should stay the same whether we are looking at Jane’s order or Jerry’s order, or whether we are looking at totals for the price or for the calories.  Similarly, we want to protect the rows 4 through 11, as they also indicate the range of items on the menu.  Conversely, we don’t want to protect row 13 or column K, as these are precisely the ones that should range freely from totals cell to totals cell.  This way, we end up with =MMULT($B13:$I13,K$4:K$11) as our formula for cell K13, and this formula can now indeed be copied and pasted into all the other cells of the totals matrix (green).

Perhaps a bit surprising, the traditional treatment of matrix multiplication in textbooks matches this latter arrangement.  When you multiply a matrix A times a matrix B, they would say, you get a matrix C with the same number of rows as matrix A and the same number of columns as matrix B.  They would say that to be able to multiply matrix A with matrix B at all, they must be conforming, which is a fancy way of saying that the number of columns of A must match the number of rows of B.  Some people like the visual image of dropping down the left matrix below the right matrix (but keeping it on the left) so that the shape of the result matrix (and the inner products that determine the value in each of its cells) can easily be seen.

Some people have never even seen any kind of visual image, and are stuck with remembering some formula they learned in college, like

How tough it must be to have any kind of real understanding of what you are doing and why, if all you learned was how to manipulate formulas like that.

The items are listed in the B column, the unit prices are listed in the C column, and Joe’s  order is in the D column.  From the amounts ordered, and the unit prices, the total for the order (ignoring taxes) is shown in the highlighted cell D12.  It shows $15.10.  Yet behind this number $15.10 is the formula that generated the number.  This formula is shown on top, just above the highlighted “D”.   The formula used is a variation of =SUMPRODUCT(C4:C11, D4:D11).  This is Excel’s version of what we’ve seen as an inner product calculation.  If you look closely at the formula, you’ll see that it really says =SUMPRODUCT($C4:$C11,D4:D11).  This dollar symbol has a special meaning to Excel, and it has nothing to do with the fact that the result of 15.10 is a dollar amount.  Rather, Excel interprets the dollar sign as protecting or freezing the column number that follows.  This is important, not for calculating the contents of the cell C12, but when we copy this formula into another cell.

In this spreadsheet, I have copied cell D12 into cell E12.  (This can be done by right-click on D12, selecting Copy, then right-click on E12, selecting Paste.  Alternatively, I can hit Control-C in D12, and Control-V in E12.  Yet another way is to drag the bottom right hand corner of the box around D12 and extend it into E12.)  Excel’s way of copying formulas is clever, in that it assumes you don’t want another identical calculation resulting in $15.10, but that you want the calculation applied to a different set of numbers.  If I had typed in =SUMPRODUCT(C4:C11,D4:D11) in cell D12 and then copied it into cell E12, it would have landed there as =SUMPRODUCT(D4:D11,E4:E11).  This is not quite what I want.  What I want is for cell E12 to be ready to give the total price for Jane’s order.  Instead, =SUMPRODUCT(D4:D11,E4:E11) would calculate the inner product of Joe’s order, and Jane’s order, a calculation for which I have no use.  What I want is for Excel to automatically modify the D4:D11 in =SUMPRODUCT(C4:C11,D4:D11) to E4:E11, but to leave the C4:C11 alone.  That is what the dollar symbol lets me express: =SUMPRODUCT($C4:$C11,D4:D11) tells Excel to leave the C’s alone, but the D’s should be modified based on where the formula is copied into.

In the same way you protect/freeze columns in formulas, you can also protect rows.  Here is a simple example, showing currency conversion.

The cell B18 contains the all-important currency conversion rate, here 1.05812 Canadian dollars for 1 US dollar.  The total amounts for Joe and Jane’s orders are shown in the C column, and the D column calculates the corresponding Canadian dollar amount.  The formula in cell D19 is one I typed in; the formula in cell D20 is copied directly from cell D19.  As you can see in the box above the highlighted D, the formula I typed in is =B$18*C19.  This formula tells two things: one, the number to show in cell D19 is what you get from multiplying the numbers in B18 and C19; two, when the formula is copied into a cell below, change the C19 accordingly, but don’t change the 18 in B18.  Instead of entering =B$18*C19, I might have entered =$B$18*C19, and thus protected it from being copied into a cell anywhere, not just below.  The main point here is that Excel doesn’t know or care, and will blithely calculate what you tell it to, whether doing so makes sense or not.

The way the Canadian dollar vector (D19:D20) depends on the US dollar vector (C19:C20) is a very common and important pattern, it is a vector operation called “multiplying by a scalar”.  In this name, the word “scalar” refers to the single outside number 1.05812, the currency conversion rate.

In the next post in this series I will show an example of matrix multiplication done in Excel.