Linear Algebra
Dynamical Geometric Problem Sheet 3: Vector space and basis
1. A Puzzle: Vector Space Operations
Instructions.
The following JSP-applet construction contains a binary operation
o and a multiplication by scalars operation * in the plane V.
You may find it convenient to look at them separately - use the buttons!
You can drag by the mouse
the vectors x and y and
control the value of the real scalar c on the pink area.
Problems
Try to figure out
a) what geometrical constructions create these operations.
b) what symbolical formulae stand behind.
Finally, you may look at an animation, where the vectors move along a line
and the scalar along the real axis (unfortunately now only between 0 and 1!).
2. Coordinates with respect to a given basis
Instructions
In the following two JSP-applet constructions you see two fixed plane vectors
e and f, together with their linear
combination ce + df.
You can control the values of the real scalars c and d
by dragging the points on the real line, situated on the pink area.
Problem 1
1. Find out the values of the coordinates of the vectors u and v.
For u:
c =
d =
For v:
c =
d =
Problem 2
1. Find out the values of the coordinates of the vectors u and v.
For u:
c =
d =
For v:
c =
d =
Martti Pesonen