Basic vector algebra
Richard G. Brereton · Journal of Chemometrics · 2016
Whether matrices or vectors are viewed as very advanced or esoteric concepts in experimental science or a straightforward way of thinking that is obvious depends on educational background. Whereas most chemometrics experts feel it is obvious that one should think in terms of vectors and matrices, many scientists do not think naturally this way. Whether one does very much depends on early education. For example, very few people feel it is a natural way of solving simultaneous equations using matrices, and most use traditional approaches of multiplication and elimination. At a more senior level, for example, even many experienced quantum chemists will not have heard of the concept of the pseudoinverse of a matrix – if you are in a University, knock on the door of a quantum chemist and find out. The introduction to the idea of vectors and matrices was quite early in a student's mathematical education many years ago; this author remembers going on an experimental course when 13 years old and being taught about eigenvalues. Modern mathematical methods of teaching were arriving, and this was proposed as an easy way of thinking – However, this needs to be introduced early to become natural. In England, over the recent years the concepts of matrices and vectors have now become less visible in the syllabus. The standard maths GCSE (a qualification usually taken at age 16 years) usually does not contain any requirement to understand about vectors and matrices. A further (or additional) maths qualification does include an introduction to “matrices – addition, subtraction and multiplication” and “determinants and solutions of linear equations” restricted to 2 × 2 matrices, and an introduction to vectors but under the subject of mechanics.1 Only when we get to A level (a qualification obtained typically at the age of 18) do we get a more detailed mention of matrices, still restricted to 3 × 3, with eigenvalues, non-singular matrices, and diagonalization,2 considered as a very small part of pure maths, and a small introduction to vector multiplication, viewed again as a tool in mechanics. Matrices are rarely taught as an applied technique, and so most school students would rarely relate these to their everyday life at this stage, and vectors appear relevant just to mechanics. Matrices and vectors are unlikely to be part of a statistics course at this stage and are regarded as a fairly abstract concept. How far students relearn about vectors and matrices in University depends on their discipline. A well-established text3 still in use since the 1970s contains 1 chapter on matrices and 1 on vectors, out of 27 chapters, but oriented toward techniques that might be most useful for physicists or engineers. Applied scientists such as analytical chemists or biologists rarely see the need for this apparently specialist approach. General analytical chemistry texts, even if there are chapters on statistics, give little or no discussion about matrices. Yet the concepts of matrices and vectors are on the whole very straightforward and can simplify thinking, but whether they are used as an everyday tool relates a bit to how confident the scientist is. Whether matrices or vectors are viewed as very advanced or esoteric concepts in experimental science (something left to advanced engineers and physicists) or a straightforward way of thinking that is obvious depends on the educational background of students or scientists. In this article we will discuss the main properties of vectors. Geometrically, the product of 2 vectors can also be defined in terms of their cosine and magnitudes so that a b = |a | |b|cos θab. We will discuss the magnitude of vectors below. We illustrate this by the example of Figure 4. One vector is expressed as a column vector and the other a row vector. Note an important misconception propagated in chemometrics that the cosine between vectors equals their correlation coefficient. We have discussed the issue of centring in the last article,5 and this property is only true if the vectors are centred. In the case of Figure 4, the correlation coefficient between the vectors is −1, and so not equal to their cosine. There is no formal definition of vector division, although some environments like Matlab do use this notation to mean something different. If you do use Matlab, be very careful what you are trying to do under such circumstances. In chemometrics there are lots of confusions about the effect of centring. For example, people often say that the cosine between 2 vectors is the same as the correlation coefficient, which is only so if the vectors are centred, but use the uncentred magnitude as an indicator of a vector's size.