• Revision:MFP4 AQA Mathematics Current Spec Alevel

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Notes for FP4, Fun stuff :|

  • 2 matrices can be added or subtracted if they have the same order. To add or subtract, just add or subtract the corresponding elements.
  • Any matrix can be multiplied by a scalar. Just multiply each element by the scalar.
  • Matrices are multiplied by multiplying the elements in a row of the first matrix by the elements in a column of the second matrix, and adding the results.
  • Commutativity of addition : For any 2 matrices which can be added, i.e. have the same order, A + B = B + A
  • Non commutativity of multiplication : It cannot be assumed that BA = AB
  • Distributive law: A(B+C) = AB + AC (U+V)W = UW + VW
  • Associative law : A(BC) = (AB)C
  • A + 0 = A, 0B = 0, C0 = 0
  • AI = A, IB = B
  • M = AB and N = BC Explain why AN = MC

Multiply the 2nd equation by A… AN = ABC as AB = M AN = MC

  • The transpose of a matrix M is obtained by interchanging the rows and columns of M. I.e. reflecting in the leading diagonal. The transpose of M is denoted by MT.
  • (AB)T = BTAT
  • Identity : all points unchanged
  • BA is the transformation A then the transformation B
  • Any linear transformation has the properties:

T(λa) = λT(a) T(a + b) = T(a) + T(b)

  • cos2θ = cos2θ – sin2θ sin2θ = 2sinθcosθ
  • The scalar product of 2 vectors a and b : a.b = |a||b|cosθ
  • The vector product of 2 vectors a and b : a x b = |a||b|sinθn
  • i x i = j x j = k x k = 0
  • i x j = k j x k = i k x i = j
  • j x i = -k k x j = -i i x k = -j
  • a x b = 0  a = 0 or b = 0 or a and b are parallel.
  • For any 2 vectors a and b, b x a = -a x b
  • For k and m any scalars, ka x mb = km a x b
  • The area of a triangle is 0.5|axb|
  • The area of a parallelogram is |axb|
  • a.bxc means a.(bxc) This is a scalar quantity – a scalar triple product. a x (bxc) is a vector quantity – a vector triple product.
  • a.bxc = b.cxa = c.axb
  • If three adjacent edges of a parallelepiped are represented by the vectors a, b and c then the volume of the parallelepiped is |a.bxc|
  • a.bxc = axb.c
  • The vector product is distributive over addition:

(a + b) x c = a x c + b x c

  • Prove that : a x (b+c) = a x b + a x c

r. a x (b+c) = r x a . (b+c) [interchange dot and cross] = r x a.b + r x a.c [distributivity of dot] = r.axb + r.axc [interchange dot and cross] r. a x (b+c) = r.axb + r.axc r. a x (b+c) - r.axb - r.axc = 0 r. (ax(b+c) – axb – axc) = 0

  • |M| is negative if the transformation represented by M is a reflection or involves a reflection.
  • The determinant of M is defined to be a.bxc and is written det(M) or |M|
  • Determinant of a 2x2 = (ad – bc)
  • The determinant is the scale factor of the enlargement of an area if 2x2 and a volume if 3x3
  • |M| = |MT|
  • Adding or subtracting any multiple of a row/column to another row/column doesn’t change the determinant.
  • Swapping 2 rows/columns changes the sign of the determinant.
  • Multiplying a row/column of a matrix by k multiplies the determinant by the k
  • To factorise a determinant use row/column operations to get a row/column of elements with a common factor.
  • |AB| = |A||B|
  • Volume of a cuboid : |a.bxc| Volume of a parallelepiped: |a.bxc|
  • Volume of a pyramid : 1/3|a.bxc| Volume of a tetrahedron : 1/6|a.bxc|
  • Volume of a triangular prism : 1/2|a.bxc|
  • a, b and c are coplanar  |a.bxc| = 0
  • The equation of a line through a and parallel to b can be expressed:

- parametric : r = a + tb - vector product : (r-a) x b = 0

  • Direction cosines: when a line is in Cartesian form it is simple. Take the vector at the bottom, and use the formula below. E.g. b1 is the i value.
  • The direction cosines of a line l,m and n satisfy : l2 + m2 + n2 = 1
  • Parametric equation of a plane : r = a + λb + μc

where λ and μ are parameters. r is the position vector of a general point on the plane, a is any specific point on the plane, b and c are 2 non-parallel vectors on the plane.

  • r.n = a.n where a is a point on the plane, n is a normal to the plane. r.n = d. Therefore the equation 3x + 2y – 7z = 12 (3, 2, -7) is n and 12 is d.
  • Find an equation of the plane through the points: a (1,0,2) , b (3,0,5) and c (-1, 6, 2)

Get 2 vectors : vector 1 = b-a vector 2 = c-b Point : (1, 0, 2) r = point + λ(vector 1) + μ (vector 2)

  • Find an equation of a plane through 2 lines: obtain a vector perpendicular to both lines by crossing the 2 direction vectors. Then do r.n=a.n where a is a point and n is the vector found from the cross product. Can then replace r with (x,y,z)
  • Find the line of intersection of 2 planes: find direction of the common line by crossing the 2 n vectors i.e. if 3x + y + 6z = 8 is one plane the (3, 1, 6) needs to be crossed with the other n vector. Then find any common point by setting for example z to zero and doing simultaneous equations, if this doesn’t work then set x or y to zero. Then do r = (common point) + t(n vector)
  • Find the point of intersection of a line and a plane: make the line equation into r=(3+t, -2+3t, 6t) then substitute in for x, y and z to get a value for t. Plug this t value back into the line equation to get the point of intersection.
  • Angle between line and plane: dot product the direction vector of the line with the n vector to the plane. Obtain a value for . If obtuse, do 180 -  = k then do 90 – k. If acute, do 90 - . This because we have found the angle between the perpendicular and the plane not the line and the plane.
  • To find the acute angle between 2 planes, just find the acute angle between their perpendiculars. I.e. the dot product.

  • To find the shortest distance between 2 lines:

-Find any vector AB from a point on one line to a point on the other -Find a vector n perpendicular to both lines -Calculate |AB.n| |n|

  • The inverse of A, any square matrix, is A-1 such that : AA-1 = A-1A = I
  • A square matrix without an inverse is singular. I.e. all matrices with a determinant of 0 are singular.
  • Algorithm for the inverse of a 3x3 matrix:

-|M|. If 0, stop. -Matrix of minor determinants -Sign change : + - + , - + - , + - + -Transpose -Divide by |M|

  • If A, B, C … are all matrices of the same size and they all have inverses, so does the product ABC. Also, (ABC…)-1 = …C-1B-1A-1
  • When considering a system of 3 equations, ensure that 2 planes aren’t the same or parallel. I.e. they’re the same if they’re multiples of one another; they’re parallel if one equation has the same x y and z coefficients but different constant terms.
  • When you do row operations, if it ends up with a row of zeros along the bottom then they’re inconsistent. If the row of zeros equals zero, then the planes meet at a line. If the row of zeros equals k not equal to zero, there are no solutions : prism.
  • A-1(Ar) = A-1b => r = A-1b i.e.number number number = A-1(xyz)
  • Row operations: equations can be multiplied by any non zero number, and any multiple of a row can be subtracted from or added to another row. Row operations can reduce to 2 equations in 2 unknowns and then to 1 equation in 1 unknown.
  • When a vector is a combination of other vectors they are linearly dependant, or when the determinant of the vectors together = 0.
  • The invariant points of the transformation of matrix M can be found by solving Mx=x
  • The invariant lines of the transformation with matrix M can be found by substituting x and mx or x and mx + c in the same equation of a line.
  • If v is an eigenvector then the line r = tv is an invariant line through the origin.
  • The eigenvectors of a transformation determine the directions of all the invariant lines that pass through the origin.
  • The eigenvectors for a matrix M can be found by solving |M-λI| = 0
  • Can show something is an eigenvector by solving Mv=λv
  • When you have an eigenvalue, can solve by setting Mx =0 where x is x,y,z matrix
  • A diagonal matrix is a square matrix that has all zeros except the leading diagonal. a matrix M which has eigenvalues and eigenvectors can be written as VDV-1 where V is the eigenvectors and D is the eigenvalues. They must be in the right order.
  • If M = VDV-1 then Mn = VDnV-1
  • Example: The matrix M has eigenvectors v1 and v2 with associated eigenvalues λ1 and λ2 . Find Mv where v = av1 + bv2

Answer: Mv = M (av1 + bv2) {substitute value for v in} = a(M v1) + b(M v2) {take a and b outside} = a(λ1 v1) + b(λ2 v2) { Mv = λv so replace Mv with λv} = λ1(a v1) + λ2 (bv2) {take λ1 and λ2 outside}

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