I undertook this in the hope that a little Linear Algebra (and Vector Calculus, the next module) would assist in understanding the basics, as I felt intimidated by the use of matrices and the mention of eigenvalues in papers, my desire was to at least grasp the maths in the papers.
The course itself progressed from elementary Vector algebra through to calculating Eigenvalues/vectors. Found it to be quite focused and
sometimes moving at a fast pace.
Definitely recommended for a crash but non trivial course, and possibly as a foundation to further study.
Being mathematically immature I was thankful I completed a basic Algebra and Trig course beforehand because they were certainly essential, wouldn’t recommend unless you are prepared for it with a little foundation even though it is described as a beginners course (Imperial style “beginners” that is).
Much emphasis is placed upon geometric visualization of algebraic operations – that is following the algebra through with vector transformations (usually in R 2 ) so an appreciation of the operations becomes much easier and actually quite interesting. If somebody had asked me before the course whether rotate and then shear could reversed with the same result I’d have been very puzzled. It’s great how the course shows you visually why this is not possible (before demonstrating the same algebraically, i.e. matrix multiplication is not commutative).
Geometric demonstration of the solving of simultaneous equations was also great fun, trivial perhaps, but when all you know is solving by algebraic methods (my understanding was limited to this before the course) this is a real eye opener to the power of Linear Algebra!
I very much also liked the geometric interpretation stuff such as the transformation of a vector in another basis using dot product and projections, and visualising the process, and later I found simply inverting the (basis) matrix (algebraically) with a results vector achieves the same result (in a determined system), but the most important thing is I can actually remember this and recall it visually (if not algebraically).
The lecturers are extremely enthusiastic and this really carries over. If I were at Imperial I’d definitely value David Dye as a lecturer.
Note:I did go through the course a second time six months after completing it and found I gained significantly more benefit the second time around being more comfortable with the topics and therefore managed to gain a better understanding from the course.
Additionally I found the use of YouTube resources (such as 3Blue1Brown) essential. This course standalone would have been hard not so much to complete, but to establish some kind of intuitive sense of what was happening, and without a lot of read-around and YouTube I suspect I would have forgotten most of it days or perhaps weeks after completing.
My experience of Calculators for Vector/Matrix course study
Interesting factoid: In 1990 Calculators represented approximately 50% or so of world computational power. In 2020 this is now less than 0.0001%
Statement: The most advanced Maths platform available is the HP-50G – nothing compares – discontinued but still available – significant advanced library support (all the way back to 28C/S software) over three decades. This calculator leaves the rest way behind, although the TI-89 Titanium does come close.
I used primarily the HP-28S (a thirty+ year old Calculator) through this course with occasional use of the later (1995) Casio FX-992S.
However, a calculator not necessary for this course although for me at least a Matrix equipped scientific helped me in verifying results (and with assignments), functions required really are just algebra and trig, so a $5 calculator would be totally usable.
However, when you are churning through Vectors and Matrices, using addition, multiplication, determinates, magnitude, dot product and other basic operations many times over – using a basic Scientific means having to remember basic formula (easy enough) and applying them monotonously (irritating) rather than focusing on conceptual blocks. In that case Vector/Matrix capabilities are extremely useful. Therefore a calculator with Matrix functionality will save you time but at the possible expense of over dependence.
At a minimum I would find useful:
- Vector Addition, Scalar multiplication, Dot product (many calculators do not have the dot product function).
- Basic Matrix Row Operations (swap, scale, add)
- REF/RREF functions
- Matrix Operations (Add/Multiply etc.)
- Matrix Determinate function
- Matrix Transpose function
- Matrix Invert function
- Vector Magnitude function (useful)
- Identity Matrix generation (useful, not essential)
- Eigenvalue/vector computation (not essential, only on the most advanced machines – TI-86/89/Voyager, HP-48/49/50 series)
Some of these are trivial to compute, but then become irritating processing manually on large vectors/matrices, so they are still handy to have at a touch of a key.
Assuming you are interested, in my experience advanced scientific calculators can be divided roughly into the following categories according to capabilities.
- Non-Scientific calculators (no trigonometric functions).
- Scientific Calculators without Matrix capabilities (the majority).
- Non-Graphing Models (some later Casio FX/Texas TI-36X Pro) can solve 3×3, and perhaps 4×4, have some useful Matrix operations, and in the case of the TI, Vectors, dot product and magnitude functions – unusual on a $20 machine.
- Graphing with basic matrix operations such as addition and multiplication, transpose, invert, identity and determinant. Along with row operations. Lower-end/mid-range Casio’s (9750GII) – these are often aimed at Higher Mathematics/College students.
- Those with additional features such as Vector support (dot product), (reduced) row echelon form functions, real numbers. Here we see most higher-end Casio’s (upgraded 9750GII/9860GII) and mid-range TI’s (81-85/Nspire) . Aimed much the same as (3).
- Those that have rich libraries often with Vector magnitude, Eigenvalue/Eigenvector computation facilities. (TI-86/TI-89/HP-28/HP-48/49/50G). These are aimed heavily at Undergraduate/Graduate Engineering students.
For sheer speed, the use of RPN, the stack, the SWAP key, and user definable softkey functions for common matrix operations (DET/INV/TRN/IDN etc) made the HP often 50-75% faster than its algebraic counterparts.
As for the best for Matrix work? The HP-50G in my opinion is the finest math platform on the planet bar none, it is a truly professional work of art, the display is great, and the functionality is endless. I barely touched its capabilities but found it amazingly intuitive in getting things done… the downside is the ENTER button is just that… a button… every HP enthusiast I suspect finds that an absolute betrayal! But it does give the 48GX folk a reason to stay put ))
HP-50G (2007-2015), HP-48GX (1994-2003), HP-28S (1987-1991) – excellent RPL based systems with amazing advanced Math capabilities. I have extensively used all three as well as the HP-48SX and the RPN stack system is second nature after a few hours. Matrices are fully integrated into the OS and are natural to work with. Personally the easiest, fastest most fluid to work with over long periods. Obviously considerable intellectual thought went into making this family of devices world class. The 48/50 series have built in Eigenvalue/vector routines, the 28S is more limited but still has pretty comprehensive matrix facilities (HP even offered a Vectors/Matrices solutions handbook for this device). The displays however (especially on the 48SX) leave a lot to be desired, and the later 48GX “Black” is definitely what you should look for (or a 50G).
The chief issue with the 28S for me was that the display of matrices would require more screen real-estate than available, requiring manual scrolling, even with 2×2 when the resultant elements were approximations (i.e. 0.0999998). Still, an amazing calculator that I use as my main device, even given its quirks.
However the 28S has its Achilles heels – battery door issues and keyboard issues (both due to case fabrication flaws) – so I would give that device a miss generally unless you know what you are getting into. Also I’d give the 48S and 48SX a miss primarily due to their substandard LCD’s, and the 49 series isn’t that great either.
If you want to go HP and RPN my advice is a 50G or a later 48GX with the “black” LCD’s (these however are not cheap).
Casio 9750 GII (2009-) (with upgraded OS to 2.0)/9860GII (2009-) –
The Casio Graphing OS has changed very little since its introduction in 1994, in fact most would not be able to tell much difference between a 1994 graphing model and contemporary models in terms of UI and for the most part functionality.
Vector/matrix work very cumbersome on these devices and I mean cumbersome, the UI really needs a work over. The LCD’s though are first rate (the best in my opinion) – crystal clear. Still current but later models are available with Colour.
The one advantage is that is has a neat system for elementary row operations on a matrix.. however it is not something I employed a great deal (granted it is a fundamental operation in solving systems by back-substitution).
Another advantage is the screen real estate allows visualisation of matrices without any need to scroll, this is very nice
However, this calculator needs: Seamless Vector/Matrix integration, get rid of the extremely annoying shift on Vct/Mat alpha labels, allow line delete by pressing DEL in any mode. Also I hate PAM but this has been on all Casio’s past around 1993… also assignable softkeys would be fantastic (given it already has function keys).
You will (unofficially) need to upgrade to the 9860GII OS 2.0 firmware to be able to take advantage of Vector arithmetic, bizarrely missing from the 9750 but available on the 9680GII.
Casio FX-991ES /(Plus) – a $10 used Calculator that many first class Universities (Oxford/Imperial/Edinburgh) recommend as the Calculator of choice for Maths and Engineering degrees (this has now likely been replaced on the lists by the EX variant). This is really all you need. Can even work with 3×3 Matrices and do Calculus (but not vector calculus ))) In all honesty anything more is a luxury and a convenience more than a necessity. The newer model (4×4) is the FX-991EX, also inexpensive. Note performing Vector/Matrix entry and operations on these isn’t exactly fast and can be frustrating.
Casio FX-992S – a $10 used core Algebra/Trig calculator that is more than sufficient for the course (and I suspect any quantitative degree). No dedicated Matrix or Calculus functions forces you to manually perform matrix algebra, on large matrices this can be tedious but surely good practice. I – Manufactured 1995 and available to about 2010 – unusually has a 12+2 digit display. Cost me the price of a cup of coffee ($5). The original list price in the 1995 Casio Japan catalogue was 8,500 yen (then about 50 UK pounds/$85), and it was classed as a higher-end Scientific, this was the end of the 992 line (the 991 still continues).
In fact the FX-992S replacement battery cost almost as much as the calculator.
TI-86 (1998-2003)/TI-89 Titanium (1999 v1 -2014 v2)/TI Nspire (2007-) – much use of these machines – solid reliable and Matrix facilities include Eigen computations. The Nspire has superb Matrix/Vector editing facilities probably the best of any device currently available (although the HP’s are more flexible in this regard). Also note the TI-89’s font is tiny and (unlike the HP-50G) fixed. Recommended if you do not wish to use RPN and calculate algebraically. I once enumerated the functions on the 86 and 89 and I think the 86 had more (but has no CAS).
Sharp EL-9900 (2000-2010) Unusual (reversible keyboard) higher-end Sharp Platform – 64K RAM and Vector/Matrix capabilities. $15 used, built like a tank literally.. has a nice weighty feel about it and for a year 2000 Calculator display is clear and easy to read. No vector magnitude function, nor dot product. In my opinion similar functions to a mid-range Casio of today. Quite nice to use as Matrix manipulation ease-of-use is quite high, easier to work with than the Casios.