Year 1 - Core Pure Flashcards
2.3 What does mod(z1z2) equal?
mod(z1z2) = mod(z1) mod(z2)
2.3 What does arg(z1z2) equal?
arg(z1z2) = arg(z1) + arg(z2)
2.3 What does mod(z1/z2) equal?
mod(z1/z2) = mod(z1)/mod(z2)
2.3 What does arg(z1/z2) equal?
arg(z1/z2) = arg(z1) - arg (z2)
3.1 What is the formula of the sum of the first n integers?
Σ(r=1 -> n) r = ((1/2)n)(n+1)
3.2 What is the formula for the sum of the first n squares?
Σ (r=1 -> n) r^2 = ((1/6)n)(n+1)(2n+1)
3.2 What is the formula for the sum of the first n cubes?
((1/4)n^2)(n+1)^2
3.1 Σ(r=1 -> n) (ar + b) = ?
aΣ(r=1 -> n)r + bΣ(r=1 -> n)1
4.1 What do the sum of 2 roots of a quadratic equal?
α + β = -b/a
4.1 What do the product of 2 roots of a quadratic equal?
αβ = c/a
4.2 What do the sum of 3 roots of a cubic equal?
α + β + γ = -b/a
4.2 What do the sum of all 3 of the pairs of products of a cubic equal?
αβ + αγ + βγ = c/a
4.2 What do the product of all 3 roots of a cubic equal?
αβγ = -d/a
4.3 What do the sum of 4 roots of a quartic equal?
α + β + γ + δ = -b/a
4.3 What do the sum of all 6 pairs of a quartic equal?
αβ + αγ + αδ + βγ + βδ + γδ = c/a
4.3 What do the sum of all 4 triples of a quartic equal?
αβγ + αβδ + αγδ + βγδ = -d/a
4.3 What does the product of all 4 roots of a quartic equal?
αβγδ = e/a
4.4 Give the rules for the sums of the reciprocals of the roots of a polynomial (quadratic, cubic, quartic)
Quadratic: (1/α) + (1/β) = (α + β)/αβ
Cubic: (1/α) + (1/β) + (1/γ) = (α + β + γ)/αβγ
Quartic: (1/α) + (1/β) + (1/γ) + (1/δ) = (α + β + γ + δ)/αβγδ
4.4 Give the rules for the products of powers of the roots of a polynomial (quadratic, cubic, quartic)
Quadratic: (α^n)(β^n) = (αβ)^n
Cubic: (α^n)(β^n)(γ^n) = (αβγ)^n
Quartic: (α^n)(β^n)(γ^n)(δ^n) = (αβγδ)^n
4.4 Give the rules of the sums of squares of the roots of a polynomial (quadratic, cubic, quartic)
Quadratic: α^2 + β^2 = (α + β)^2 - 2αβ
Cubic: α^2 + β^2 + γ^2 = (α + β + γ)^2 - 2(αβ + αγ + βγ)
Quartic: α^2 + β^2 + γ^2 + δ^2 = (α + β + γ + δ)^2 -2(αβ + αγ + αδ + βγ + βδ + γδ)
4.4 Give the rules of the sums of cubes of the roots of a polynomial (quadratic, cubic)
Quadratic: α^3 + β^3 = (α + β)^3 - 3αβ(α + β)
Cubic: α^3 + β^3 + γ^3 = (α + β + γ)^3 - 3(α + β + γ)(αβ + αγ + βγ) + 3αβγ
4.5 e.g. The cubic x^3 - 2x^2 + 3x - 4 = 0 has roots α, β, & γ, find the equation with roots 2α, 2β, and 2γ
Let w = 2x
x = w/2
Substitute x = w/2 into the cubic, simplify until coefficients are integers
7.1 What can any linear transformation be represented by?
A matrix
7.1 What matrix can the linear transformation T:(x over y)->(ax + by over cx + dy)?
T=(TL(a), TR(b), BL(c), BR(d))
7.2 What is an invariant line?
A line that doesn’t move under a transformation
7.2 What matrix represents the linear transformation of a reflection in the y-axis? Which points will be the invariant points? Which lines will be the invariant lines?
(TL(-1), BL(0), TR(0), BR(1))
Invariant points are all the points along the y-axis
Invariant lines include x=0, and y=k (for any value of k)
7.2 What matrix represents the linear transformation of a reflection in the x-axis? Which points will be the invariant points? Which lines will be the invariant lines?
(TL(1), BL(0), TR(0), BR(-1))
Invariant points are all the points along the x-axis
Invariant lines include y=0, and x=k (for any value of k)
7.2 What matrix represents the linear transformation of a reflection in the line y=x? Which points will be the invariant points? Which lines will be the invariant lines?
(TL(0), BL(1), TR(1), BR(0))
Points on the line y=x are invariant points
Invariant lines include y=x and y=-x+k (for any value of k)
7.2 What matrix represents the linear transformation of a reflection in the line y=-x? Which points will be the invariant points? Which lines will be the invariant lines?
(TL(0), BL(-1), TR(-1), BR(0))
Points on the line y=-x are invariant points
Invariant lines include y=-x and y=x+k (for any value of k)
7.2 What matrix represents the linear transformation of a rotation through angle θ clockwise about the origin? Which points will be the invariant points? Which lines will be the invariant lines?
(TL(cosθ), BL(sinθ), TR(-sinθ), BR(cosθ))
Only invariant point is the origin
For θ≠180°, there are no invariant lines. For θ=180°, any line passing through the origin is invariant.
7.3 What matrix represents a stretch/enlargement?
(TL(a), BL(0), TR(0), BR(b)) is a stretch with scale factor a in the x-direction and scale factor b in the y-direction
In the case where a=b, it is an enlargement with scale factor a
7.3 What are the invariant lines for the linear transformation of a stretch/enlargement in both directions?
x-axis and y-axis are invariant lines
Origin is an invariant point
7.3 For a stretch parallel to the x-axis only, what are the invariant points and invariant lines?
- Points on the y-axis are invariant points
- Any line parallel to the x-axis is an invariant line
7.3 For a stretch parallel to the y-axis only, what are the invariant points and invariant lines?
- Points on the x-axis are invariant points
- Any line parallel to the y-axis is an invariant line
7.3 For a linear transformation represented by 𝐌, what does det(𝐌) represent? What is this sometimes called?
det(𝐌) represents the scale factor for the change in area, sometimes called the area scale factor
7.4 When considering two linear transformations represented by matrices P and Q, what does the matrix PQ represent?
The transformation Q, with matrix Q, followed by the transformation P, with matrix P
7.5 What matrix represents a 3D transformation of a reflection in the plane x=0?
(TL(-1), TM(0), TR(0), ML(0), MM(1), MR(0), BL(0), BM(0), BR(1))
7.5 What matrix represents a 3D transformation of a reflection in the plane y=0?
(TL(1), TM(0), TR(0), ML(0), MM(-1), MR(0), BL(0), BM(0), BR(1))
7.5 What matrix represents a 3D transformation of a reflection in the plane z=0?
(TL(1), TM(0), TR(0), ML(0), MM(1), MR(0), BL(0), BM(0), BR(-1))
7.5 What matrix represents a 3D transformation of a rotation of angle θ about the x-axis?
(TL(1), TM(0), TR(0), ML(0), MM(cosθ), MR(-sinθ), BL(0), BM(sinθ), BR(cosθ))
7.5 What matrix represents a 3D transformation of a rotation of angle θ about the y-axis?
(TL(cosθ), TM(0), TR(sinθ), ML(0), MM(1), MR(0), BL(-sinθ), BM(0), BR(cosθ))
7.5 What matrix represents a 3D transformation of a rotation of angle θ about the z-axis?
(TL(cosθ), TM(-sinθ), TR(0), ML(sinθ), MM(cosθ), MR(0), BL(0), BM(0), BR(1))
7.6 What linear transformation is represented by the matrix A^(-1)?
The reversal of the transformation described by matrix A
7.6 What linear transformations are self-inverse?
Reflections
8.1 Describe the process for proof by induction
- Basis - Prove the general statement is true for n=1
- Assumption - Assume the general statement is true for n=k
- Inductive - Show the general statement is then true for n=k+1
- Conclusion - General statement is then true for all positive integers n
8.1 What are the sentences to remember for proof by induction?
- When n=1
- therefore true for n=1 as RHS=LHS
- Assume true for n=k
- When n=k+1
- Hence true for n=k+1
- Since true for n=1, if true for n=k, then true for n=k+1, therefore true for all n
8.2 How do you use proof by induction to prove divisibility results?
- Show n=1 is true
- Assume n=k is true: show f(k)
- Show n=k+1 is true: show f(k+1)
- Manipulate f(k+1) to = f(k) + a(x) where d is the multiple you are trying to show
8.3 How do you use proof by induction with matrices?
- Show n=1 is true
- Assume n=k is true: show M^k
- Show n=k+1 is true: multiply M^k by M
9.1 What is the vector equation of a straight line passing through the point A with position vector 𝐚, and parallel to the vector 𝐛?
𝐫 = 𝐚 + λ𝐛
9.1 What is the cartesian form of a vector equation?
If 𝐚 = (𝐚1 𝐚2 𝐚3) and 𝐚 = (𝐛1 𝐛2 𝐛3), it can be written as (x-𝐚1)/𝐛1 = (y-𝐚2)/𝐛2 = (z-𝐚3)/𝐛3
9.2 What is the vector equation of a plane in 3D?
𝐫 = 𝐚 + λ𝐛 + μ𝐜
where 𝐫 is the position vector of a general point on the plane, 𝐚 is the position vector of a point on the plane, 𝐛 and 𝐜 are non-parallel non-zero vectors in the plane, and λ and μ are scalars
9.2 What is the cartesian form for the equation of a plane in 3D?
ax + by + cz = d
where a, b, c, and d are constants, and (a over b over c) is the normal vector to the plane
9.3 What is the formula for the angle between two vectors?
θ = cos^(-1) ((𝐚.𝐛)/mod(𝐚)mod(𝐛))
9.3 What indicates that two vectors are perpendicular in terms of scalar products?
𝐚.𝐛 = 0
9.3 What indicates that two vectors are parallel in terms of scalar products?
𝐚.𝐛 = mod(𝐚)mod(𝐛)
9.3 What is 𝐚.𝐛?
𝐚.𝐛 = (𝐚1 over 𝐚2 over 𝐚3).(𝐛1 over 𝐛2 over 𝐛3) = 𝐚1𝐛1 + 𝐚2𝐛2 + 𝐚3𝐛3
9.4 What is the scalar product form of the equation of a plane?
𝐫.𝐧 = 𝐤
where 𝐤 = 𝐚.𝐧 for any point in the plane with position vector 𝐚
9.4 How do you find the angle between two planes?
Work out the angle between the normals of the planes, subtract this from 180 to find the angles between the planes
9.5 How do you determine whether two lines meet?
- Write the equations in vector notations and set them equal to each other
- Write three linear equations involving λ and μ
- Try to solve the first two equations simultaneously
- If there are no solutions, the lines do not meet
- Do these solutions satisfy the third equation?
- If yes, the lines intersect
- If no, the lines do not intersect
9.5 What does it mean if two straight lines are skew?
They are not parallel and do not intersect
9.6 How do you find the shortest distance between two non-intersecting lines?
Find the line segment AB such that A lies on l1 and B lies on l2 and AB is perpendicular to both lines
9.6 How do you find the shortest distance between a point P and a plane Π?
Find the length of the line segment that goes through point P and is parallel to the normal vector of the plane Π
9.6 How do you find the shortest distance between a line l and a point P?
Find the length of the line segment that is perpendicular to l and goes through P
9.6 What is the equation involving the length of the perpendicular from the origin to a plane Π and the equation of a plane written as 𝐫.𝐧 = 𝐚.𝐧?
𝐫.𝐧̂ = k
where 𝐧̂ is a unit vector perpendicular to Π and k is the length of the perpendicular from the plane Π to the origin
9.6 What is the equation for the perpendicular distance from the point with coordinates (α, β, γ) to the plane with equation ax + by + cz = d?
mod(aα + bβ + cγ - d)/sqrt((a^2) + (b^2) + (c^2))
