Following on from the group effort PoSWW, this is a group effort WitCH. It is worth keeping the PoSWW in mind, since the zero vector is again a primary source of confusion and error.
The focus is again Year 12 Specialist texts, this time on their coverage of linear dependence and independence. We have already WitCHed this in reference to the previous edition of Cambridge (which remains essentially unchanged on this topic). Readers may feel free to cheat by referring to that WitCH. Although Cambridge’s treatment was and is atrocious, however, the main targets are Nelson and Jacaranda.
CAMBRIDGE
NELSON
JACARANDA
The Nelson attempt is just plain wrong, so I will concentrate on what Jacaranda gives, where the issue is a little more subtle. The Jacaranda first sentence is true as a logical statement but fails as a {\em definition} of linear dependence. If there are non-zero
, 
, 
where 
then 
, 
and 
are indeed linearly dependent. But there are other situations where the vectors are linearly dependent, for example where some, but not all of 
, 
, 
, are zero.
This reminds me of lectures by the great mathematician and greater teacher, E. R. Love. When making a definition, he would write using the format “xxxxx means yyyyyyy”. This practice requires that yyyyyyy does include all cases where the definition applies. Most textbooks are still deficient in this regard.
Thanks, tom. I like Love’s “x means y” formulation. Nelson is, of course, hilarious (but not all wrong). I think there’s plenty more one could say about Jacaranda.
Jacaranda’s definition of linear dependence requires all of the scalars to be non-zero and for linear independence they must all be zero. Which means that there are sets of vectors that are neither linearly dependent nor independent.
Yeah. Maybe a third gender or something.
Nicely explained anon. My point is that the first 2 paras can be interpreted as an example, not as a definition, of linear dependence. So it can be argued that there is no actual error, just bad teaching. However, the bold type for the words “linear dependent” does make it smell like a definition. Anyway, I am on a campaign to get authors to use the word “means” (a la Love) to remove any doubt.
Love had a wide reputation for precision in his publications. I’m guessing this was learned when he worked as a proof-reader for the “austere” G. H. Hardy.
https://www.austms.org.au/wp-content/uploads/Gazette/1998/Mar98/love.html
I’ll start with a swipe at Jacaranda:
A set of vectors is linearly dependent if it is not linearly independent. A set is therefore linearly independent if it fails the test for linear dependence.
Was that so hard? OK, my definition is not perfect, but I’m not being paid to write or review this stuff.
Nelson: I really want to make a joke about it being “iffy” – but I’ll leave the jokes to Marty as he is much, much better at them.
Nelson seemed to start off OK, with the first part of the first sentence. There was another good line in there a bit later but so much waffle at best confuses the situation and at worst makes it wrong.
Cambridge: 0 is a real number. I’ll leave it there.
“iffy” was pretty good.
Other people have already commented on the Nelson and Jacaranda texts, and I don’t feel like looking over them right now.
The Cambridge text doesn’t seem too bad, but there’s one small hangup I have. The book doesn’t mention how to check for linear independence for a set of three vectors in the case that all three vectors are parallel, nor does it explicitly mention anywhere the consequence that any set of vectors where *any* two vectors are parallel is linearly dependent. Ideally the latter would be mentioned somewhere (e.g. in the “Note:” above), and the check should be labelled as being for three non-parallel vectors.
For that matter, since a and b are assumed to be nonzero and non-parallel, {a,b} is linearly independent. Thus we can (and probably should) extend that check to the more general statement that if {v_1, v2_, v_3, …, v_n} is a linearly independent set of vectors, then {v_1, v_2, v_3, …, v_n, v_(n+1)} is linearly independent if and only if v_(n+1) is not a linear combination of {v_1, v2_, v_3, …, v_n}. Then the book could mention this statement explicitly, and give the check as a special case for n = 2.
Cambridge is appalling. It only looks good when put next to the Nelson and Jacaranda junk.
Halmos (FDVS, p. 7): “A finite set
of vectors is linearly dependent if there exists a corresponding set 
of scalars, not all zero, such that 
.
If, on the other hand,
implies that 
for each 
, the set 
is linearly independent.”
Halmos notes that this definition covers the case when the set of vectors is empty (p. 8).
Cambridge said (roughly) this.
Unfortunately, Cambridge also said lots of other things on the same page.
The “not all zero” for me is the key bit.
But then, if the set contains a zero vector, this can have any scalar as the coefficient, make the coefficients of all non-zero vectors zero and… you have a linearly dependent set, do you not?
I don’t have access to any of these books, but I have a terrible feeling that they all launch straight into this abstract idea of linear dependence. At school level surely a teacher would work up to it to make the whole thing relevant, and visual. Maybe first discuss how every 3D vector can be written as a combination of
, 
and 
. Then, is it always possible to write every 3D vector as a combination of 3 given vectors? Only if they are not co-planar. All this can be done with diagrams and minimal algebra. Then investigate what co-planar means algebraically.
More generally, it seems that teachers usually teach a topic the way they have been taught. If they first met the topic at university and then find themselves teaching it in secondary schools, we get this result.
It is bad at Victorian universities. It is much, much worse in Victorian schools.
I don’t know how controversial this opinion may be, but I think they should only teach about vectors in the context of mechanics in VCE, and since they mechanics entirely, not at all. It seems like they’re just teaching linear (in)dependence for the sake of teaching it. If they want to teach it, they should teach the entirety of linear algebra, and since that’s too much for high school students, not at all.
And to pick some low hanging fruit, according to Cambridge, if you reorder a list of three linearly dependent vectors, the list need not be linearly dependent anymore.
Thanks, A. I don’t think that opinion is the slightest bit controversial, and it is certainly the only sane stance.
Which line in Cambridge implies the reordering thing?
The purple section titled “Linear Dependence for three vectors.”
Nevermind, I misread it. Oops.
Not all your fault …
To go back to your first point, the density of Cambridge and the errordom of the other two is compelling evidence that no one reads any of it, that no one gives a stuff about this topic other than to survive it. It is madness.
Right, I’ve done a horrific job of writing this, but at least I managed to inexplicably introduce the term “iff”. Finally, the students will be educated!
If I might chime in late here … the grave error made by all texts is that pointed out initially by Tom, that “But there are other situations where the vectors are linearly dependent, for example where some, but not all of
, are zero.” The correct definition is given by Halmos as quoted by Terry Mills. But the biggest error really is that somehow the texts seem to infer that linear independence is equivalent to writing a specific vector as a linear combination of the others. An easy counterexample is a = [1,0], b = [2,0], c = [0,1], where c cannot be written as a linear combination of a and b. The point is that linear independence is a set property: a property of the entire set of vectors. And all texts seem to hiccup over the difference of “not all zero” and “all not zero” which are two very different beasts indeed.
Both Nelson and Jacaranda are simply wrong in their very first sentence – their attempt at a definition; and in Jacaranda’s case this is exacerbated by a truly horrible choice of symbols. According to Nelson, the set of vectors above would be linearly independent. Jacaranda puts their (its?) foot in it in the very first sentence, and puts the other foot in with the incorrect inference given in their second paragraph. A true paradise of errors.
Cambridge alone considers a general case of an arbitrary number of vectors; it seems at least “less wrong” than the others, with the confusion, already pointed out by Red Five, of zero, and “real numbers”. But it seems pretty murky; I’ve no idea how befuddled students (and their long-suffering teachers) would manage with it.
The question one might ask is: why get it so wrong, when with a modicum of care it would be just as easy to get it right?
I don’t know whether to be embarrassed, horrified, or furious.
Thanks, Alsadair. Very good summary of the underlying causes of the wonkiness. As for the reason they can get it so wrong: because the topic is one little twig of pointlessness. There’s no general discussion of or need to understand the algebra of vector spaces, of span and so forth. It’s just one, lonely pair of concepts, to be applied in only the generic setting, where an utter lack of understanding cannot derail the required mechanical response. The blame, as always, lies with VCAA.
Unfortunately – most teachers look to past exams to determine what VCAA thinks is important enough to test and how to test it.
Linear dependence of a set of vectors always seems to be in 3-dimensional space and (in my limited experience) seems to either be a multiple choice question (where, as we know, there is no guarantee any option will be correct) or a stand-alone, couple of marks early on Paper 1 where the questions tend to be pretty routine.
To be honest, this is not a part of the course that causes me to lose much sleep.
Of course you don’t lose much sleep, because, rightly, your only concern is that students can do the exam questions. On the other hand, if one is concerned with whether anybody in VCE doing these questions has absolutely any clue what they’re doing, or why, …
Maybe I have just reached the point of acceptance that VCAA is not going to change and that my time is better spent working out how to navigate a semi-sensical path through the swamp.
It does bother me greatly that the textbooks/exams/whatever that students should be able to assume are going to give them accurate information are failing in this rather fundamental duty. The people who pay my wages only care about results though.
Of course the primary responsibility of any teacher is to do the best they can within the status quo. Do teachers also have an ethical/professional obligation to fight against the status quo? It’s arguable, but it’s not obvious. But there is a much larger problem than inertia: the problem of ignorance.
The longer we go on, the higher the percentage of teachers who were taught within this fundamentally anti-intellectual system, and thus have absolutely no sense that they are simply reproducing the nonsense they were themselves provided. Forgive them, a little, for they know not what they do, at all.
Of course the issue of why this topic should be taught at all is a big issue. I can’t see why it would be needed at VCE level. It’s quite a sophisticated concept, and I remember initially struggling with it on my initial exposure in my first year uni algebra subject. But surely, one can reasonably ask, if any topic, for whatever reason, is to be included in what is laughingly referred to a syllabus, then at least it can be done right.
“Surely, we can be reasonable”, pleaded Alice, as the March Hare swatted the Mad Hatter with the Dormouse.
Seagoon: Stop, or I’ll horsewhip you with this!
Eccles: Put me down!