Algebra Basics

Recognise that algebra is a fundamental skill at A-level. It is assumed that you are competent, even perfect, at the GCSE algebra skills. This particularly includes:
Simultaneous Equations (and a check, too)
Quadratics; factorising, solving, formula method (big equation), completing the square, sketching
Trigonometry; including the sine and cosine rules (an extension of Pythagoras’ Theorem). Treat trig as algebra (it becomes harder)
Transformations ; yes this is in Algebra, because the ideas belong here and in work you do with functions.

Simultaneous Equations
1) Graphical methods: quick, dirty (not precise)
2) Elimination method:
3) Substitution method: works for non-linear equations too
4) Matrix method: takes the same length of time irrespective of the complexity


Transformations     (the full six)            ( x )
Translation - vector representation      ( y )
Rotation - of angle in direction about a centre
Enlargement - of scale factor about centr
Reflection - about an axis
Stretch - parallel to lines (two way) an
d scale factors
Shear - invariant line required


An  Algebra  is a set of symbols (elements) and rules for their manipulation.
There are roughly four categories of algebra:

    * Elementary algebra, in which the properties of operations on the real number system are recorded using symbols (as place-holders) to denote constants and variables, and we study the rules governing mathematical expressions and equations involving these symbols.
    * Abstract (modern) algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated. Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. This tends to be left to the far (optional) end of Further Pure, but the idea is worth a look.
    *
Linear algebra, in which the specific properties of vector spaces are studied (including matrices). Note especially that with matrices, in general AB≠BA, so multiplication is not commutative (though addition is). Which is why I introduce them early if you let me.
    *
Universal algebra, in which properties common to all algebraic structures are studied.


In advanced studies, axiomatic algebraic systems such as groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. This tends to be beyond school study, but is quite readable.


Abstract Algebra  - the essential ideas
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of objects called elements. All the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax² + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.
Binary operations: The notion of addition (+) is abstracted to give a binary operation, * say. For two elements a and b in a set S a*b gives another element in the set, (technically this condition is called closure). Addition (+), subtraction (-), multiplication (×), and division (÷) are all binary operations as in addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are abstracted to give the notion of an identity element. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator * the identity element e must satisfy a * e = a and e * a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. However, if we take the positive natural numbers and addition, there is no identity element.

Inverse elements: The negative numbers gives rise to the concept of an inverse elements. For addition, the inverse of a is -a, and for multiplication the inverse is 1/a. A general inverse element a-1 must satisfy the property that a * a-1 = e and a-1 * a = e.

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2+3)+4=2+(3+4). In general, this becomes (a * b) * c = a * (b * c). This property is shared by most binary operations, but not subtraction or division. Confusion with this leads to difficulty with negatives, see the year Eight page

Commutativity: Addition of integers also has a property called commutativity. That is, the order of the numbers to be added does not affect the sum. For example: 2+3=3+2. In general, this becomes a * b = b * a. Only some binary operations have this property. It holds for the integers with addition and multiplication, but it does not hold for matrix multiplication. Lower school students get stuck on this: is 3x4 really the same as 4x3 ? The fundamental confusion lies with what they think the numbers stand for (they don’t, they’re just numbers)

© David Scoins 2017