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Maths Techiniques (Relations (Reflexive - Every element in A is in B,…
Maths Techiniques
Relations
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Order Relation - Ref, Anti-Sym and Trans
Equivalence Relation - Ref, Sym and Trans
Relation - Given sets A, B, C etc.. R is a subset of A x B x C
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Equivalence Class - Given an equivalence relation A, with element a, an equivalence class is all values which are equivalent to a
Given a and b, if b ∈ [a]≈ then a≈b else a≈!b
Vectors
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Vector - P = p + v. Eg P = (2, 3) + v(4, -2)
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Intersections in 3D - Equate the two lines/planes and write 3 equations in each dimension. Solve using Gaussian Elimination
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Gaussian Elimination
Rules - Can add or subtract two rows, or multiply an entire row by a constant or swap two rows
Echelon form - Stair case of 0's, with diagonal values above
Enhanced echelon form - Diagonal of values, everything else 0's
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Contradictory Equation - Eg. x + 2y = 3 and 2x + 4y = 7. Contradict each other so no solution. In Gaussian, one size is all 0's
Irrelevant Equation - Both sizes are 0's. This means that it doesn't constrain the value of x and it can be chosen freely
Co-efficient is 0 - If the co-efficient is 0 for that row, then swap it with another row.
More Vectors
Normal - P=sv+tw+c -> N = vw OR ax + by = c -> n = (a, b)
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Laws of Inner Product
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<v + w, x> = <v, x> + <w, x>
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Dot/Inner Product - v = (v1, v2, v3). <v, w> = v1w1, v2w2, v3w3
If <v, w> = 0 then right angles. |v| x |w| x cos(a)
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Distance From plane - (d - <n, Q>)/|n| or <n, P-Q>/|n|
Nearest Neighbour - Q' = Q + (d-<n,Q>)/<n,n> .n
Sets
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Pairing - AxB elements are pairs (x, y)
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Cardinality
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Countable - If it has the same number of elements as N and they can be paired together. Eg Z and N^2
Uncountable - Cannot be paired to N. Eg. R, I