Hyperplanes

Vectors of field type real numbers are difficult to visualize if n is not 1,2, or 3. Familiar objects like lines and planes make sense for any value of n. Line L along the direction defined by a vector v, through a point P labeled by a vector u, can be written as follows:

L = {u + tv | t ∈ R}

Given two non-zero vectors, u and v, they determine a plane if both the vectors are not in the same line, and one of the vectors is a scalar multiple of the other. The addition of two vectors is accomplished by laying the vectors head to tail in a sequence to create a triangle. If u and v lie in a plane, then their sum lies in the plane of u and v. The plane represented by two vectors u and v can be mathematically shown as follows:

{P + su + tv | s, t ∈ R}

We can generalize the notion of a plane as a set of x + 1 vectors and P, v1, . . . , vx in R, n with x ≤ n determines a x-dimensional hyperplane:

(P + X x i=1 λivi | λi ∈ R)