Problem solving and proving via generalisation
... The third example shows the value and power of problem solving via general pattern or structure rather than just carrying out calculations by brute force. The next example demonstrates the utility of seeing a sequence as a function with the set of natural numbers as domain, and using algebraic techn ...
... The third example shows the value and power of problem solving via general pattern or structure rather than just carrying out calculations by brute force. The next example demonstrates the utility of seeing a sequence as a function with the set of natural numbers as domain, and using algebraic techn ...
Complex numbers 1 Complex algebra and geometry
... 14. |eiθ | = 1 for all real θ, so these complex numbers lie on the unit circle in the complex plane. 15. eiπ + 1 = 0, a remarkable equation, involving 0, 1, e, π and i. 16. Polar form of a complex number: z = reiθ , where r and θ are the polar coordinates of z on the complex plane. The relation to t ...
... 14. |eiθ | = 1 for all real θ, so these complex numbers lie on the unit circle in the complex plane. 15. eiπ + 1 = 0, a remarkable equation, involving 0, 1, e, π and i. 16. Polar form of a complex number: z = reiθ , where r and θ are the polar coordinates of z on the complex plane. The relation to t ...
Matrix multiplication and composition of linear
... Theorem 2 Let T : Rp → Rn be any function. Then the following are equivalent: 1. There exists an n × p matrix B such that T = TB , i.e., such that T (X) = TB (X) for all X ∈ Rp . 2. T satisfies the principle(s) of superposition: (a) T (X + X 0 ) = T (X) + T (X 0 ) for all X and X 0 in Rp , and (b) ...
... Theorem 2 Let T : Rp → Rn be any function. Then the following are equivalent: 1. There exists an n × p matrix B such that T = TB , i.e., such that T (X) = TB (X) for all X ∈ Rp . 2. T satisfies the principle(s) of superposition: (a) T (X + X 0 ) = T (X) + T (X 0 ) for all X and X 0 in Rp , and (b) ...
