• k - flats

    Let \[\mathbb{R}^n\] be the set of n - tuples. A k - flat \[F\] , or linear variety or affine subspace, in \[\mathbb{R}^n\] is the set of all n - tuples of the form \[P+V\] , where \[P\] is the position vector of a point of \[\mathbb{R}^n\] and...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5056-k-flats.html

  • Linear Transformations With Quaternions

    The set of all quaternions \[q=a+b \mathbf{i}+c \mathbf{j}+d \mathbf{k}\] where \[a, \; b, \; c, d \in \mathbb{R}\] with \[\mathbf{i}^2= \mathbf{j}^2 = \mathbf{k}^2 =-1\] \[ij=k, \; jk=i, \; ki=j, \; ji=-k, \;kj=-i, \; ik=-j\] forms a vector space over...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5065-linear-transformations-with-quaternions.html

  • Modelling Inbreeding With Matrices

    In the brother system problem, a male and a female mate, and from their direct descendants, two individuals of opposite sex are selected at random. Direct offspring of these two are mated and the process continues. The possible genotype for the parents...

    https://astarmathsandphysics.com/university-maths-notes/matrices-and-linear-algebra/5089-modelling-inbreeding-with-matrices.html

  • Testing a Game For Bias Example

    Two players pay dice with the following rule. If a player tosses a 6, he wins. If a 4 or 5 is tossed the player throws the dice again. If a 1, 2 or 3 is tossed, the dice passes to the other player. Is the game biased? We can construct the table: A's...

    https://astarmathsandphysics.com/university-maths-notes/game-theory/5098-testing-a-game-for-bias-example.html

  • Properties of Division

    Theorem (Properties if Division) Let \[a, \; b\] be positive integers, and let \[c, \; d\] be any integers. Then a) If \[a | c\] then \[a | (c+na)\] for any integer \[n\] . b) If \[c \neq 0\] and \[a | c\] then \[a \le |c|\] . c) If \[a | b\] and \[b |...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5128-properties-of-division.html

  • Proof of Form of Divisors of Mersenne Numbers

    Theorem Any prime divisor of the Mersenne Primes \[M_p=2^p-1\] where \[p\] is an odd prime, is off the form \[2kp+1\] for some positive integer \[k\] Proof Let \[q\] be a prime divisor of \[M_p=2^p-1\] . Then \[2^p \equiv 1 \; (mod \; q)\] . From...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5154-proof-of-form-of-divisors-of-mersenne-numbers.html

  • Euler's Theorem

    Theorem: If \[a \in \mathbb{Z}_n^*\] then \[a^{\phi(n)} = 1 \pmod{n}\] . This reduces to Fermat’s Little Theorem when \[n\] is prime. Proof: Let \[m = \phi(n)\] , and label the units \[u_1,...,u_m\] . Consider the sequence \[a u_1,...,a u_m\] (we...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/4323-euler-s-theorem.html

  • Diophantine Equation With No Solutions

    Theorem (Diophantine Equation With No Solutions) The equation \[x^4+y^4=z^2\] has no integer solutions. Proof Suppose \[x_1, \; y_1, \; z_1 \] is a solution. Suppose \[gcd(x, \; y)=d \gt 1\] then \[x^4+y^4=z^2 \rightarrow (dx')^4+(yd')^4=z^2 \rightarrow...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5199-diophantine-equation-with-no-solutions.html

  • Combining Coins to Make £6

    Using only 2 pence, 10 pence and 50 pence coins, in exactly how many ways can 100 coins be made to total £6? We can write the problem as the pair of simultaneous equations \[2x+10y+50z=600 \rightarrow x+5y+25z=300\] \[x+y+z=100\] where \[x, \; y, \; z\]...

    https://astarmathsandphysics.com/university-maths-notes/number-theory/5219-combining-coins-to-make-6.html

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