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Physical effects involve things acting on other things to produce a change of position, tension etc. These effects usually depend upon the strength, angle of contact, separation etc of the interacting things rather than on any absolute reference frame so it is useful to describe the rules that govern the interactions in terms of the relative positions and lengths of the interacting things rather than in terms of any fixed viewpoint or coordinate system. Vectors were introduced in physics to allow such relative descriptions.

The use of vectors in elementary physics often avoids any real understanding of what they are. They are a new concept, as unique as numbers themselves, which have been related to the rest of mathematics and geometry by a series of formulae such as linear combinations, scalar products etc.

Vectors are defined as "directed line segments" which means they are lines drawn in a particular direction. The introduction of time as a geometric entity means that this definition of a vector is rather archaic, a better definition might be that a vector is information arranged as a continuous succession of points in space and time. Vectors have length and direction, the direction being from earlier to later.

Vectors are represented by lines terminated with arrow symbols to show the direction. A point that moves from the left to the right for about three centimetres can be represented as:

------------------->

If a vector is represented within a coordinate system it has components along each of the axes of the system. These components do not normally start at the origin of the coordinate system. 

The vector represented by the bold arrow has components a, b and c which are lengths on the coordinate axes. If the vector starts at the origin the components become simply the coordinates of the end point of the vector and the vector is known as the position vector of the end point.

If two vectors are connected so that the end point of one is the start of the next the sum of the two vectors is defined as a third vector drawn from the start of the first to the end of the second:

c is the sum of a and b:

If the components of a are a, b, c and the components of b are d, e, f then the components of the sum of the two vectors are (a+d), (b+e) and (c+f). In other words, when vectors are added it is the components that add numerically rather than the lengths of the vectors themselves.

1. Commutativity a + b = b + a

2. Associativity (a + b) + c = a + (b + c)

If the zero vector (which has no length) is labelled as 0

3. a + (-a) = 0

4. a + 0 = a

The discussion of components and vector addition shows that if vector a has components a,b,c then qa has components qa, qb, qc. The meaning of vector multiplication is shown below:

The bottom vector c is added three times which is equivalent to multiplying it by 3.

1. Distributive laws q(a + b) = qa + qb and (q + p)a = qa + pa

2. Associativity q(pa) = qpa

Also 1 a = a

If the rules of vector addition and multiplication by a scalar apply to a set of elements they are said to define a vector space.

An element of the form:

q₁a₁ + q₂a₂ + q₃a₃ +.... + q_ma_m

is called a linear combination of the vectors.

The set of vectors multiplied by scalars in a linear combination is called the span of the vectors. The word span is used because the scalars (q) can have any value - which means that any point in the subset of the vector space defined by the span can contain a vector derived from it.

Suppose there were a set of vectors (a₁,a₂,.... ,a_m) , if it is possible to express one of these vectors in terms of the others, using any linear combination, then the set is said to be linearly dependent. If it is not possible to express any one of the vectors in terms of the others, using any linear combination, it is said to be linearly independent.

In other words, if there are values of the scalars such that:

(1). a₁ = q₂a₂ + q₃a₃ +.... + q_ma_m

the set is said to be linearly dependent.

There is a way of determining linear dependence. From (1) it can be seen that if q₁ is set to minus one then:

q₁a₁ + q₂a₂ + q₃a₃ +.... + q_ma_m = 0

So in general, if a linear combination can be written that sums to a zero vector then the set of vectors (a₁,a₂,.... ,a_m) are not linearly independent.

If two vectors are linearly dependent then they lie along the same line (wherever a and b lie on the line, scalars can be found to produce a linear combination which is a zero vector). If three vectors are linearly dependent they lie on the same line or on a plane (collinear or coplanar).

If n+1 vectors in a vector space are linearly dependent then n vectors are linearly independent and the space is said to have a dimension of n. The set of n vectors is said to be the basis of the vector space.

Also known as the 'dot product' or 'inner product'. The scalar product is a way of removing the problem of angular measures from the relationship between vectors and, as Weyl put it, a way of comparing the lengths of vectors that are arbitrarily inclined to each other.

Consider two vectors with a common origin:

The projection of a on b is:

P = | a | cos q

Where | a | is the length of a.

The scalar product is defined as:

(2) a . b = | a | | b | cos q

(2) also allows cos q to be defined as:

cos q = a . b / ( | a | | b |)

The definition of the scalar product also allows a definition of the length of a vector in terms of the concept of a vector itself. The scalar product of a vector with itself is:

a . a = | a | | a | cos 0

cos 0 (the cosine of zero) is one so:

a . a = a²

which is our first direct relationship between vectors and scalars. This can be expressed as:

(3) a = (a . a)^1/2

where a is the length of a.

Properties:

1. Linearity [Ga + Hb].c = Ga.c + Hb.c

2. symmetry a.b = b.a

3. Positive definiteness a.a is greater than or equal to 0

4. Distributivity for vector addition (a + b).c = a.c + b.c

5. Schwarz inequality | a.b | £ ab

6. Parallelogram equality | a + b |² + | a - b |² = 2( | a |² + | b |²)

7. a.b = a₁b₁ + a₂b₂ + a₃b₃

This gives us the length of a vector in terms of coordinates (Pythagoras' theorem) from:

8. a.a = a² = (a₁) ² + (a₂) ² + (a₃)²

The derivation of 7 is:

a = a₁i + a₂j + a₃k

where i, j, k are unit vectors along the coordinate axes. From (4)

a.b = (a₁i + a₂j + a₃k) .b = a₁i.b + a₂j.b + a₃k.b

but b = b₁i + b₂j + b₃k

so:

a.b = b₁a₁i .i + b₂a₁i .j + b₃a₁i .k + b₁a₂j.i + b₂a₂j.j + b₃a₂j.k + b₁a₃k.i + b₂a₃k.j + b₃a₃k.k

i .j, i .k, j .k, etc. are all zero because the vectors are orthogonal, also i .i, j.j and k.k are all one (these are unit vectors defined to be 1 unit in length).

Using these results:

a.b = a₁b₁ + a₂b₂ + a₃b₃

Matrices are sets of numbers arranged in a rectangular array. They are especially important in linear algebra because they can be used to represent the elements of linear equations.

11a + 2b = c

5a + 7b = d

The constants in the equation above can be represented as a matrix:

            11    2

A =

            5     7

The elements of matrices are usually denoted symbolically using lower case letters:

            a₁₁    a₁₂

A =

            a₂₁    a₂₂

Matrices are said to be equal if all of the corresponding elements are equal.

Eg: if a_ij= b_ij

Then A = B

Matrices are added by adding the individual elements of one matrix to the corresponding elements of the other matrix.

c_ij= a_ij + b_ij

or C = A + B

Matrix addition has the following properties:

1. Commutativity A + B = B + A

2. Associativity (A + B) + C = A + (B + C)

and

3. A + (-A) = 0

4. A + 0 = A

From matrix addition it can be seen that the product of a matrix A and a number p is simply pA where every element of the matrix is multiplied individually by p.

Transpose of a Matrix

A matrix is transposed when the rows and columns are interchanged:

        a₁₁ a₁₂ a₁₃

A = a₂₁ a₂₂ a₂₃

        a₃₁ a₃₂ a₃₃

          a₁₁ a₂₁ a₃₁

A^T = a₁₂ a₂₂ a₃₂

          a₁₃ a₂₃ a₃₃

Notice that the principal diagonal elements stay the same after transposition.

A matrix is symmetric if it is equal to its transpose eg: a_kj = a_jk.

It is skew symmetric if A^T = -A eg: a_kj = -a_jk. The principal diagonal of a skew symmetric matrix is composed of elements that are zero.

Diagonal matrix: all elements above and below the principal diagonal are zero.

    4  0  0

    0 -1  0

    0  0  2

Unit matrix: denoted by I, is a diagonal matrix where all elements of the principal diagonal are 1.

    1  0  0

    0  1  0

    0  0  1

Matrix multiplication is defined in terms of the problem of determining the coefficients in linear transformations.

Consider a set of linear transformations between 2 coordinate systems that share a common origin and are related to each other by a rotation of the coordinate axes.

Two Coordinate Systems Rotated Relative to Each Other

If there are 3 coordinate systems, x, y, and z these can be transformed from one to another:

x_{1 =} a₁₁y₁ + a₁₂y₂

x_{2 =} a₂₁y₁ + a₂₂y₂

y_{1 =} b₁₁z₁ + b₁₂z₂

y_{2 =} b₂₁z₁ + b₂₂z₂

x_{1 =} c₁₁z₁ + c₁₂z₂

x_{2 =} c₂₁z₁ + c₂₂z₂

By substitution:

x_{1 =} a₁₁(b₁₁z₁ + b₁₂z₂) + a₁₂(b₂₁z₁ + b₂₂z₂)

x_{2 =} a₂₁(b₁₁z₁ + b₁₂z₂) + a₂₂(b₂₁z₁ + b₂₂z₂)

x_{1 =} (a₁₁b₁₁ + a₁₂b₂₁)z₁ + (a₁₁b₁₂ + a₁₂b₂₂)z₂

x_{2 =} (a₂₁b₁₁ + a₂₂b₂₁)z₁ + (a₂₁b₁₂ + a₂₂b₂₂)z₂

Therefore:

c_{11 =} (a₁₁b₁₁ + a₁₂b₂₁) c₁₂ = (a₁₁b₁₂ + a₁₂b₂₂)

c_{21 =} (a₂₁b₁₁ + a₂₂b₂₁) c₂₂ = (a₂₁b₁₂ + a₂₂b₂₂)

The coefficient matrices are:

        a₁₁      a₁₂

A =  a₂₁      a₂₂

        b₁₁      b₁₂

B =  b₂₁      b₂₂

        c₁₁     c₁₂

C =  c₂₁     c₂₂

From the linear transformation the product of A and B is defined as:

                                (a₁₁b₁₁ + a₁₂b₂₁)          (a₁₁b₁₂ + a₁₂b₂₂)

C = AB =

In the discussion of scalar products it was shown that, for a plane the scalar product is calculated as: a.b = a₁b₁ + a₂b₂ where a and b are the coordinates of the vectors a and b.

Now mathematicians use a sleight of hand that probably deserves closer inspection; they define the rows and columns of a matrix as vectors:

                                      b₁₁

A Column vector is b = b₂₁

And a Row vector a = a₁₁        a₁₂

Matrices can be described as vectors eg:

         a₁₁   a₁₂           a₁

A =   a₂₁   a₂₂    =    a₂

and

       b₁₁    b₁₂

B = b₂₁    b₂₂ = b₁     b₂

Matrix multiplication is then defined as the scalar products of the vectors so that:

C = a₂.b₁      a₂.b₂

From the definition of the scalar product a₁.b₁ = a₁₁b₁₁ + a₁₂b₂₁ etc.

In the general case:

       a₁.b₁     a₁.b₂    .    a₁.b_n

C = a₂.b₁     a₂.b_{2     .}     a₂.b_n

This is described as the multiplication of rows into columns (eg: row vectors into column vectors). The first matrix must have the same number of columns as there are rows in the second matrix or the multiplication is undefined.

After matrix multiplication the product matrix has the same number of rows as the first matrix and columns as the second matrix:

1 3 4 times 2    has 2 rows and 1 column    39

6 3 2          3                                                35

                  7

1. Not commutative AB ¹ BA

2. Associative A(BC) = (AB)C

(kA)B = k(AB) = A(kB)

3. Distributative for matrix addition

(A + B)C = AC + BC

matrix multiplication is not commutative so C(A + B) = CA + CB is a separate case.

4. The cancellation law is not always true:

AB = 0 does not mean A=0 or B=0

There is a case where matrix multiplication is commutative. This involves the scalar matrix where the values of the principle diagonal are all equal. Eg:

       k  0  0

S = 0   k  0

       0  0  k

In this case AS = SA = kA. If the scalar matrix is the unit matrix: AI = IA = A.

A simple linear transformation such as:

x_{1 =} a₁₁y₁ + a₁₂y₂

x_{2 =} a₂₁y₁ + a₂₂y₂

can be expressed as:

eg:

x₁ a₁₁ a₁₂ y₁

= *

x₂ a₂₁ a₂₂ y₂

and

y_{1 =} b₁₁z₁ + b₁₂z₂

y_{2 =} b₂₁z₁ + b₂₂z₂

as: y = Bz

Using the associative law:

and so:

C = AB =

Consider a simple rotation of coordinates:

x^m is x₁ , x₂

xⁿ is x'₁ , x'₂

The scalar product can be written as:

s.s =g_{m n} x^m xⁿ

Where:

            1       0

g_{m n} =

            0        1

and is called the metric for this 2D space.

s.s = g₁₁ x₁x'₁ + g₁₂ x₁x'₂ + g₂₁ x₂x'₁ + g₂₂ x₂x'₂

Now, g₁₁ = 1, g₁₂ = 0, g₂₁ = 0, g₂₂ = 1 so:

s.s = x₁x'₁ + x₂x'₂

If there is no rotation of coordinates the scalar product is:

s.s = x₁x₁ + x₂x₂

s² = x₁² + x₂²

Which is Pythagoras' theorem.

Indexes that appear as both subscripts and superscripts are summed over.

g_{m n} x^m xⁿ = g₁₁ x₁x'₁ + g₁₂ x₁x'₂ + g₂₁ x₂x'₁ + g₂₂ x₂x'₂

g_m ⁿ x^m xⁿ = g₁₁ x₁x'₁ + g₂₁ x₂x'₁ where n = 1

by promoting n to a superscript it is taken out of the summation.

Consider:

Columns times rows:

x₁ times y₁   y₂ = x₁ y₁       x₁ y₂

x₂                         x₂ y₁       x₂ y₂

Matrix product XY = x_iy_j Where i = 1, 2 j = 1, 2

There being no summation the indexes are both subscripts.

 Rows times columns:

x₁     x₂ times y₁ = x₁ y₁ + x₂ y₂

                       y₂

Matrix product XY = d _ijxⁱy^j

Where d _ij is known as Kronecker delta and has the value 0 when i ¹ j and 1 when i = j. It is the indicial equivalent of the unit matrix:

1 0

0 1

There being summation one value of i is a subscript and the other a superscript.

A matrix in general can be specified by any of:

M_i^j , M_ij , M^ij , Mⁱ_j depending on which subscript or superscript is being summed over.

A vector can be expressed as a sum of basis vectors.

x = a₁e₁ + a₂e₂ + a₃e₃

In indicial notation this is: x = aⁱe_i

Consider x = Ay where A is a coefficient matrix and x and y are coordinate matrices.

In indicial notation this is:

x^m = A^m _n xⁿ

which becomes:

x₁ = a₁₁ x'₁+ a₁₂ x'₂+ a₁₃ x'₃

x₂ = a₂₁ x'₁+ a₂₂ x'₂+ a₂₃ x'₃

x₃ = a₃₁ x'₁+ a₃₂ x'₂+ a₃₃ x'₃

In indicial notation the scalar product is:

x.y = d _ijxⁱy^j

In matrix notation the scalar product of two column vectors is:

        x₁                  y₁

X =  x₂         Y = y₂

        x₃                  y₃

x.y = X^TY = d _ijxⁱy^j

Important classes of second order differential equations are examples of the "Sturm-Liouville equation". These consist of the differential equation plus boundary conditions over some interval. A typical example is:

d²y/dx² + λy = 0

For positive values of λ the solution is obtained by substituting λ=ν² so that

A can be found to be zero from the first boundary condition where y(0) = 0, therefore:

Now λ=ν² so λ= 1,4,9 etc.. (ν²=0 results in y = 0 which is not a permitted solution)

When B equals 1:

The function sin νx is known as a characteristic function, or eigenfunction, of this problem and the characteristic values, or eigenvalues, are the values of λ.

In other words an eigenfunction is a solution of the equation that satisfies the boundary conditions and the eigenvalues are values of the constant in the main "Sturm-Liouville equation" equation that apply in this circumstance. These characteristic properties are very important when considering wave equations. In the case of a vibrating string the eigenvalues specify resonant modes of vibration, a value of 1 being a half cycle, 2 a full cycle and so on, because the boundary values incorporate no displacement at the ends where the string is fixed.

These have many uses, including solving simultaneous differential equations ie: equations such as:

d²y₁/dt² = -5y₁ + 2y₂

d²y₂/dt² = 2y₁ - 2y₂

This sort of equation is common in the description of combinations of harmonic motions. Mathematicians solve this sort of equation by converting it into a "vector equation"

The right hand side, -5y₁ + 2y₂ and 2y₁ - 2y₂ is expressed as two matrices:

And the second differential coefficients are expressed as:

So the overall vector equation is Y = Ay (those familiar with vector multiplication should be able to easily reconstitute the original simultaneous equations see above).

The solution of second order differential equations is usually accomplished by substituting an exponential function, in this case this is achieved by substituting

This means that the solution of the simultaneous equations must conform to solutions of Ax = λx. Remarkably, this equation can be solved if the values that compose A are known. If x and λ can be found then, from y = xe^ωt, y can be found at any t and the equation solved. The vector, x, is known as an eigenvector of A and the constant λ is known as an eigenvalue of A. How x and λ are found from A is discussed below.

This vector equation can be written out in full as:

Which becomes, transferring the right hand side to the left becomes Bx = 0 ie:

This set of simultaneous linear equations is solved using determinants. In this particular case, where there are n linear equations in n unknowns all with a solution of zero (ie: homogenous) the equations only have non-trivial solutions if the determinant of the coefficients is zero.

This determinant is zero: D(λ) = 0 and is known as the characteristic determinant.

Returning to the problem of solving Ax = λx. Suppose A has the following values:

Resolving the determinant:

D(λ) = λ ² - 7 λ + 6 = 0 

This equation is known as the characteristic polynomial. The roots in this case are: λ₁ = 6 and λ₂ = 1. Evaluating Bx = 0 gives:

-x₁ + 4x₂ = 0

x₁ - 4x₂ = 0

so x₁ = 4x₂

The eigenvector of A, x₁, is, using the eigenvalue from λ₁ = 6:

Another eigenvector of A, x₂, is, using the eigenvalue from λ₂ = 1:

The values of the eigenvectors and eigenvalues can be substituted into the original equations, such as Y = Ay, to allow them to be solved.