Basic Mathematics Notes, Study notes of Mathematics

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Basic Mathematics

Contents

1 Basic Skills

This document contains notes on basic mathematics. There are links to the corresponding Leeds

University Library skills@Leeds page, in which there are subject notes, videos and examples.

If you require more in-depth explanations of these concepts, you can visit the Wolfram Math-

world website:

⇒ Wolfram link (http://mathworld.wolfram.com/ )

• Algebra (Expanding brackets, Factorising) :

⇒ Library link

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/algebra/ ).

• Fractions :

⇒ Library link

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/numeracy/fractions.html).

• Indices and Powers :

⇒ Library link

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/numeracy/indices.html).

• Vectors :

⇒ Library link

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/mechanics/vectors.html).

• Trigonometry and geometry :

⇒ Library link

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/trig geom/ ).

• Differentiation and Integration :

⇒ Library link

(http://library.leeds.ac.uk/tutorials/maths-solutions/pages/calculus/ ).

1.1 Practice Questions

There are practice equations available online to accompany these notes.

2 Linear Algebra

⇒ Wolfram link (http://mathworld.wolfram.com/LinearAlgebra.html)

2.1 Matrices and Vectors

⇒ Library link (http://library.leeds.ac.uk/tutorials/maths-solutions/pages/mechanics/vectors.html)

2.1.1 Definitions

A matrix is a rectangular array of numbers enclosed in brackets. These numbers are called

entries or elements of the matrix.

e.g.

A =

Matrix A has 2 rows and 3 columns.

A row vector is a matrix with a single row:

e.g. (^) [

]

Whereas a column vector is a matrix with a single column:

e.g. 

The size of a matrix is defined by n × m where n is the number of rows and m is the number

of columns. Matrix A, as defined in equation 1, is a 2 × 3 matrix.

An element of a matrix can be described by its row position and column position. For ex-

then,

A + B =

a 11 + b 11 a 12 + b 12

a 21 + b 21 a 22 + b 22

2.1.4 Subtraction

Similar to addition, corresponding elements in A and B are subtracted from each other:

A − B =

a 11 − b 11 a 12 − b 12

a 21 − b 21 a 22 − b 22

2.1.5 Multiplication by a scalar

If λ is a number (i.e. a scalar) and A is a matrix, then λA is also a matrix with entries

λa 11 λa 12

λa 21 λa 22

2.1.6 Multiplication of two matrices

⇒ Wolfram link (http://mathworld.wolfram.com/MatrixMultiplication.html)

This is non-trivial and is governed by a special rule. Two matrices A , where A is of size

n × m, and B of size p × q, can only be multiplied if m = p, i.e. the number of columns in

A must match the number of rows in B. The matrix produced has size n × q, with each entry

being the dot (or scalar) product (see section 2.1.10) of a whole row in A by a whole column in

B.

e.g. if

A =

 and^ B^ =

then

AB =

(1 × 1) (1 × 2) (1 × 3)

+ (3 × 5) + (3 × 6) + (3 × 7)

+ (6 × 9) + (6 × 10) + (6 × 11)

(0 × 1) (0 × 2) (0 × 3)

+ (− 1 × 5) + (− 1 × 6) + (− 1 × 7)

+ (4 × 9) + (4 × 10) + (4 × 11)

Formally, if

AB = C then

cij =

∑^ m

k=

aikbkj (9)

Aside

When using Matlab (or octave), two matrices can be multiplied in an element-wise sense. This

is NOT the same as described above.

1. How many bolts are needed for Alex’s car parts?

(0 × 3) + (1 × 1) + (4 × 2) = 9.

1. How many screws are needed for Peter’s car parts?

(1 × 4) + (0 × 8) + (2 × 4) = 6.

Or we can use matrix multiplication to get all four scenarios:

AB =

2.1.8 Matrix-vector multiplication

Since a vector is a special case of a matrix, this is simply a special case of the matrix-matrix

multiplication we have already discussed. Consider multiplying a column vector of length m

by a matrix of size n × m,

e.g.

which results in a column vector of length n and in this case n = 2.

2.1.9 Special Matrices

Identity Matrix, I: The identity matrix, I, of size n × n, is defined in equation 12.

aij =

1 if i = j

0 if i 6 = j

i.e. if n = 2

I =

This is a special case of a diagonal matrix possessing non-zero entries only on its diagonal e.g.

If A is a square n × n matrix, then the identity matrix In×n has the special property that:

AI = IA = A (14)

NB: I is the equivalent of 1 in scalar arithmetic i.e. 1 × 2 = 2 × 1 = 2.

Transpose, A

T : If A is a n × m matrix then the transpose of A, denoted A

T , is a m × n

matrix found by swapping rows and columns of A,

e.g.

if A =

 A

T

Inverse matrix, A

− 1 If A is an n×n matrix, sometimes (see later) there exists another matrix

called the inverse of A, written A

− 1 , such that

AA

− 1 = A

− 1 A = I (15)

NB: For scalar numbers, x

− 1 is the inverse of x when considering multiplication, since

xx

− 1 = x

− 1 x = 1 (16)

2.2 Linear Systems

⇒ Wolfram link (http://mathworld.wolfram.com/LinearSystemofEquations.html)

⇒ Video link (http://www.youtube.com/watch?v=ZK3O402wf1c)

A linear system of equations such as

5 x + 3y + z = 3

2 x − 10 y − 3 z = − 1

4 y + 5z = 7

can be written as (^) 

x

y

z

as can be verified by multiplying out the left hand side. When solving the linear system Ax = b,

(where A is a matrix, x is the vector (x, y, z,... ) and b is a vector of numbers) two cases can

arise:

i) A

− 1 exists.

Then A

− 1 Ax = A

− 1 b ⇒ Ix = A

− 1 b

so x = A

− 1 b

ii) A

− 1 doesn’t exist. There is then no solution in general.

2.3 Determinants

⇒ Wolfram link (http://mathworld.wolfram.com/Determinant.html)

⇒ Video link (http://www.youtube.com/watch?v=23LLB9mNJvc)

How do we know when A

− 1 exists? One method is to calculate the determinant of A, writ-

ten det A or | A |. The determinant is a single number that contains enough information about

A to determine whether it is invertible.

2 ×2 determinants: In the 2 × 2 case, if

A =

a b

c d

then | A |= ad − bc. For example, for the linear system given by

x + 2y = 2

3 x + 4y = 3

the determinant of the coefficient matrix is

The matrix A is therefore invertible (see section 2.3.1) and so the solution exists.

As another example, consider

x + 2y = 2

2 x + 4y = 3

or (^) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 0 0

Do whichever is easier!

2.3.1 Using determinants to invert a 2 × 2 matrix

The determinant can be used in finding the inverse of a 2 × 2 matrix.

For A =

a b

c d

, the inverse can be found using the formula

A

− 1

detA

d −b

−c a

 =^

ad − bc

d −b

−c a

For example: find the inverse of matrix B =

B

− 1

− 1

3 2

1 2

2.4 Eigenvalues and Eigenvectors

⇒ Wolfram link 1 (http://mathworld.wolfram.com/Eigenvalue.html)

⇒ Wolfram link 2 (http://mathworld.wolfram.com/Eigenvector.html)

⇒ Video link (http://www.youtube.com/watch?v=lXNXrLcoerU)

Often we are interested in whether a matrix can stretch a vector. In such a case:

Av = λv , (21)

where λ is the “stretch factor”. The scalar λ is called an eigenvalue (from the German: eigen

meaning same) and v is an eigenvector. Av = λv is equivalent to:

(A − λI)v = 0. (22)

If det(A − λI) 6 = 0 then the system can be solved to find v = (A − λI)

− 1 0 = 0. If we want

non-zero vectors v, then we require |A − λI| = 0.

To find the eigenvectors and eigenvalues, we use a two stage process:

i) Solve |A − λI| = 0 ,

ii) Find v.

For example:

A =

i) The eigenvalues λ are such that

 −^ λ

For example: if λ = 4 above, then

1 1

2 2

4 4

are all eigenvectors. We typically choose the

simplest!

Matrix diagonalisation Suppose A is a n × n matrix. Form a new matrix V , each column of

which is an eigenvector of A. If V

− 1 exists then,

V

− 1 AV = Λ or A = V ΛV

− 1 where Λ is a diagonal matrix with (λ 1 , λ 2 ,... ),

the eigenvalues of A on the diagonal.

E.g.

A =

 then^ V^ =

 ,^ Λ =

NB - Eigenvector 1 is:

1 1

, eigenvector 2 is:

2 1

It turns out that V

− 1 exists and is:

V

− 1

Check:^ V V^

− 1

So then the matrix product

V

− 1 AV =

is equal to the matrix Λ.

Special case: Symmetric matrix A matrix A is symmetric if A = A

T

. It turns out that, in

this case, all the eigenvectors are orthogonal, i.e. if V 1 and V 2 are different eigenvectors, then

V 1 · V 2 = 0.

If each eigenvector is normalised such that Vi · Vi = 1, then,

V

T = V

− 1 (This is an easy way to find V

− 1 ).

Example:

A =

 (A

T = A)

Eigenvectors λ are given by:

(1 − λ)

2 − 4 = 0

λ

2 − 2 λ − 3 = 0 or (λ − 3)(λ + 1) = 0

so, λ = 3 or λ = − 1

λ = − 1 : 

x

y

 ⇒^ x^ +^ y^ = 0

Hence, eigenvector is:

1 − 1

But this can be normalised (since it’s magnitude is arbitrary) by any number μ : μ

1 − 1

Let’s choose μ such that: v

T v = 1, where v = μ

1 − 1

⇒ 1 = μ

2

 = 2μ

2 .