FM - Unit 1 Flashcards
give it a minute
Proof by induction
- Base case
- Assume n=k is true
- Rewrite the n=k+1 case in terms of the n=k case
For divisibility, use ℕm in a substitution in the n=k+1 case
For series, the n=k+1 case is identical to n=k but with an extra term
For matrices, use power laws
Thus it has been proven by induction that since the base case is true and the n=k case is assumedly true, then it must follow that the n=k+1 case is true, implying that the statement is true for n E N
Loci - Circle
|z - a|= r
Center is at “a” and radius is “r”
Loci - Half-line
arg(z - a) = θ
Origin is at “a” and anticlockwise angle from r-axis is “θ”
Loci - Perpendicular Bisector
|z - a| = |z - b|
Where the line that joins “a” and “b” is the line’s perpendicular bisector.
Loci - Region
|z - a| > |z - b|
Where the perpendicular bisector separates the space into two regions, signifies the “less than” side.
If in doubt geometrically…
…draw a triangle
Sum Of r
ⁿΣᵣ₌₁(r) = (n(n+1)) / 2
Sum of r²
ⁿΣᵣ₌₁(r²) = (n(n+1)(n+1/2)) / 3
Sum of r³
ⁿΣᵣ₌₁(r³) = (n²(n+1)²) / 4
Summation / Method of Differences
A series is expressed as a difference between two other series. When expanded manually (helps to visualise in a matrix) many terms cancel, allowing for simpler simplification.
Summation / Method of Seperation
A series can be split into two or more series that can be simplified.
Sum Of First n Odd Numbers
ⁿ⁻¹Σᵣ₌₀(2r+1)
Summation Algebra Rules
Can take out constants & iterate it over addition
Matrix Determinant
det(A) = ad - bc
Represents the area scale factor that the matrix A will provide.
Matrix Inverse
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 |
Matrix Rotation
(cosθ -sinθ \ sinθ cosθ)
To turn a matrix by angle θ around the origin anticlockwise.
Matrix Reflection
(cos(2θ) sin(2θ) \ sin(2θ) -cos(2θ))
To reflect in the line y = (tanθ)x
Non-Linear Matrix Transformation
Where T(a b) = (a + x b + y)
T = (1 0 x \ 0 1 y / 0 0 1)
Angle Between Planes And Vectors
Between two planes or vectors
cosθ = (a⋅b) / (|a||b|)
Between a plane and vector
sinθ = (a⋅n) / (|a||n|)
To Find Equation of Plane
n ⋅ a = 0
n ⋅ b = 0
a,b are parallel to the plane
n is the normal
Roots
-b/a = sum
c/a = sum of pair products
-d/a = sum of triplet products
. . .
for f(nx), roots of f(x) are divided by n to get roots of f(nx)
Conditions To Operate Matrices
Addition/subtraction - Must be the same dimensions
Multiplication - Number of columns must match the number of rows
Singular Matrix
det(M) = 0
Matrix Invariant Lines
M(x mx + c) = (x y)
substitute (x y) into mx + y and match coefficients
Linear vs Non-Linear Transformations
A linear transformation always preserves a linear relationship between the variables, while a non-linear transformation distorts it.
Complex Number Parameters Operations
Multiply r
Add theta
