FM - Unit 1 Flashcards

give it a minute

1
Q

Proof by induction

A
  1. Base case
  2. Assume n=k is true
  3. Rewrite the n=k+1 case in terms of the n=k case

For divisibility, let the n=k case equal ℕm
For series, the the n=k+1 case is identical to n=k but with an extra term
For matrices, use power laws

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2
Q

Loci - Circle

A

|z - a|= r

Center is at “a” and radius is “r”

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3
Q

Loci - Half-line

A

arg(z - a) = θ

Origin is at “a” and anticlockwise angle from r-axis is “θ”

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4
Q

Loci - Perpendicular Bisector

A

|z - a| = |z - b|

Where the line that joins “a” and “b” is the line’s perpendicular bisector.

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5
Q

Loci - Region

A

|z - a| > |z - b|

Where the perpendicular bisector separates the space into two regions, signifies the “less than” side.

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6
Q

If in doubt geometrically…

A

…draw a triangle

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7
Q

Sum Of r

A

ⁿΣᵣ₌₁(r) = (n(n+1)) / 2

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8
Q

Sum of r²

A

ⁿΣᵣ₌₁(r²) = (n(n+1)(n+1/2)) / 3

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9
Q

Sum of r³

A

ⁿΣᵣ₌₁(r³) = (n²(n+1)²) / 4

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10
Q

Summation / Method of Differences

A

A series is expressed as a difference between two other series. When expanded (helps to visualise in a matrix) many terms cancel, allowing for simpler simplification.

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11
Q

Summation / Method of Seperation

A

A series can be split into two or more series that can be simplified.

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12
Q

Sum Of First n Odd Numbers

A

ⁿ⁻¹Σᵣ₌₀(2r+1)

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13
Q

Summation Algebra Rules

A

Can take out constants & iterate it over addition

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14
Q

Matrix Determinant

A

det(A) = ad - bc

Represents the area scale factor that the matrix A will provide.

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15
Q

Matrix Inverse

A

A⁻¹ = 1/det(A) x (d -b \ -c a)

If det(A) = 0, the matrix has no inverse as any shape it transforms will turn into 0 area.

d | -b |

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16
Q

Matrix Rotation

A

(cosθ -sinθ \ sinθ cosθ)

To turn a matrix by angle θ around the origin anticlockwise.

17
Q

Matrix Reflection

A

(cos(2θ) sin(2θ) \ sin(2θ) -cos(2θ))

To reflect in the line y = (tanθ)x

18
Q

Non-Linear Matrix Transformation

A

Where T(a b) = (a + x b + y)
T = (1 0 x \ 0 1 y / 0 0 1)

19
Q

Angle Between Planes And Vectors

A

Between two planes or vectors
cosθ = (a⋅b) / (|a||b|)

Between a plane and vector
sinθ = (a⋅n) / (|a||n|)

20
Q

To Find Equation of Plane

A

n ⋅ a = 0
n ⋅ b = 0

a,b are parallel to the plane
n is the normal

21
Q

Reflection of Point In Plane

22
Q

Reflection of Line In Plane

23
Q

Center of Mass of Triangle

A

1/3 along perpendicular bisector from base

24
Q

Center of Mass of Semi-Circle

A

4/3pi along the radius