Dual Basis

We will have a Vector Space V and a One-form Space V*. Each is the dual space of the other. In most turorials V* is simple referred to as “the Dual Space”. We want to place a huge emphasis on the fact that V is dual to V*.

Every possible pairing of a vector v from the vector space and a value k from the field creates an element of the dual space.

The basis set spanning V will be v_1, v_2.

The basis set spanning V* will be w^1, w^2.

We will use unit vectors for the vectors in the basis set for V.

There is a math law that

f^j(v_i) = \delta^j_i

f^1(v_1) = 1 , f^1(v_2) = 0 f^2(v_2) = 1 f^2(v_2) = 1

Magic Glass Pane ($60)

We can have points that don’t have coordinates but they have locations.

It is fair for Descartes to tell us they are near-useless.

Our magic glass lets us see things, and if you are a good student of magic, you can duplicate things using a plain piece of paper, a compass and a straight edge.

Numbers aren’t bad. The problem is the evil of the coordinates. Some persons may have told you to stay away from numbers (or it sounded like they said that) because it was a simple way to guarantee “coordinate-free”.

show picture of a tornado, M5.

Magic+ Glass Pane ($599.95)

When you draw a second vector with tail-to-head joining to a first

Origin

When you impose coordinates and an origin, points get numbers.

Rational Numbers

Any rational number a/b is equal to c/d where we choose d and calculate c.

We already knew ‘a’ and ‘b’. We decided what c would be.

c = \dfrac {da}{b}

Banach Space

We can have lengths

We cannot claim a location for a single point.

Is that enough to tell us that the point is on glass and the graph paper behind the glass can be moved?