Given any vector space V over a field F, the dual space V∗ is defined as the set of all linear maps φ: V → F (linear functionals).
Source: Dual space – Wikipedia
the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph.
This course is an elementary introduction to number theory. Topics to be covered include: Primes, Divisibility and the Fundamental Theorem of Arithmetic Greatest Common Divisor (GCD), Euclidean Algorithm Congruences, Chinese Remainder Theorem, Hensel’s Lemma, Primitive Roots Quadratic Residues and Reciprocity Arithmetic Functions, Diophantine Equations, Continued Fractions,
If W is a linear subspace of a finite-dimensional vector space V, then the codimension of W in V is the difference between the dimensions:
Source: Codimension – Wikipedia
For every such rational number p/q, draw a circle Cp/q of diameter 1/q2 that touches the x-axis exactly at p/q, and sits above this x-axis.