Two-dimensional Linear Space with ordered Basis (i, j)

Lehman - Pesonen (Spring 2004, 26.2.2014, last revised 13.2.2019)

1. Ordered Basis (i, j) in the 2-dimensional plane

We take an ordered basis E := (i, j) in the Plane, that is; two non-parallel (non-collinear) vectors i and j, whose order we keep fixed.

Unit Area

In this first Sketch the ideas of Basis and Unit Area are introduced.

The parallelogram generated ('spanned') by the unit vectors is always defined, but can be of area zero. We take the area of the parallelogram as the Unit Area of value 1.

Question 1
What happens to the areas when the vectors are parallel?

2. Vectors in the two-dimensional Linear Space with ordered Basis (i, j)

We have a fixed ordered basis E := (i, j) in the Plane.
Every vector in the Plane is a linear combination of these vectors.

Basis spans the vector set

In this Sketch the basis E is 'arbitrary'. All vectors ares spanned by the two base vectors.
For a vector u, the coordinates with respect to the basis E are (a,c).

Question 2
What are the coordinate values for the vector v?

How to calculate with vectors, see the axioms in

Definition of General linear space

3. Signed area of parallelogram defined by two vectors in a two-dimensional Linear Space with ordered Basis E = (i, j)

We have a fixed ordered basis E = (i, j) in the Plane and two vectors u and v. They define a parallelogram, whose signed area we introduce now.

Signed Area

In this Sketch the basis E is 'arbitrary', but fixed. The picture is supposed to show how the Unit Area and the vector parallelogram (signed) area are tied together.

Question 3
a) What is the signed area of the yellow parallelogram OPQRO?
b) What is the signed area of the yellow parallelogram ORQPO?

4. How the determinant describes area in orthonormal (rectangular unit vectors) basis case

Euclid - Lehman - Pesonen

We have an ordered basis E := {i,j} in the Plane, and two vectors u and v with coordinates (a,c) and (b,d) in the basis E.

Determinant of two vectors

In this Sketch the basis E is given. The picture is supposed to show how the determinant ties the unit area and the vector parallelogram together.

Push the buttons Signed Area and Transform. Then drag the vector \( \mathbf{v} \), draggable on the vertical axis, until a STOP sign appears.
The same time you should see \( \mathbf{u} \) draggable and drag it to the horozontal axis.
Now you have a rectangle wth the same area as the parallelogram generated by \( \mathbf{u} \) and \( \mathbf{v} \).

Martti Pesonen 2004, 2014, 13.2.2019 (+ Henri Tanskanen)