Linear Algebra

Dynamical Problem Sheet 2: Scaling Functions

- towards the concept multiplication by scalars in linear spaces
DEAR STUDENT
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Purpose of the Worksheet
The purpose of this Worksheet is to reinforce the understanding of abstract binary operations; the special target here are the functions
(scalar,vector) |--> vector
where vectors stay in a certain basic set, ie. vector set, and scalars act (scale) from some external set.

Contents

Brief Instructions
I Definition Identification Tasks
II VSG Identification Tasks
III VSG Production Tasks

IV Student Feedback

Brief Instructions

This example applet construction visualizes the standard vector addition +, multiplication by scalars (as the scaling function) and linear combinations of plane vectors.
This page requires a Java capable browser. In these JavaSketchpad applet constructions which we call sketches here) you usually can drag (move in the usual way using the mouse) the red endpoints. These represent usually either point in the plane or real values on a line.
The red point labeled u in the sketch beside corresponds to a plane vector starting from the origin of the two-dimensional Euclidean plane R2, while the red point c tied to a line, stands for a real scalar (number) on the real line R.
The vector c*u is the image of the pair (c,u) with respect to the scaling function; we call it the result of the scaling in question.

A sketch is activated by a mouse click. To restart an active sketch from its original situation press the key 'R'.

I Scaling Functions: Definition Identification Tasks

In each of the following plane geometry figures you see a scalar c on the real line, a vector u and the result c*u of the operation.

Let us recall the definition of the concept "external binary operation" or "scaling function for a set" (in Finnish: "ulkoinen laskutoimitus" or "skaalausfunktio jollekin joukolle"):


DEFINITION OF THE SCALING FUNCTION
Let V and K be nonempty sets.
Any two variable function K×V ®V is called a scaling function for V.
This means: for every ordered pair (c,u) in K×V there must be exactly one image element in V.
The image c*u is called the result of the scaling. The scaling set K is called the scalar set and the scaled set V the vector set.
In our course the scalar set is usually the field of real numbers R, but the vectors can be whatsoever.

I Problem 1. In which of the Figures A-F the operation is a scaling function for the whole plane V := R2 ?
Maximum 6 x 1 = 6 points.

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Figure A
Figure B
Figure C
Is Figure A a scaling function for R2? yes no I don't know
Is Figure B a scaling function for R2? yes no I don't know
Is Figure C a scaling function for R2? yes no I don't know
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Figure D
Figure E
Figure F
Is Figure D a scaling function for R2? yes no I don't know
Is Figure E a scaling function for R2? yes no I don't know
Is Figure F a scaling function for R2? yes no I don't know
I Problem 2. In the case(s) where the operation is not a scaling function for R2, explain why it is not.
Maximum 4 points.

Your answer to I Problem 2:

II Scaling Functions: VSG Identification Tasks

In what follows you see three kinds of mathematical expressions: Verbal, Symbolical and Graphical (VSG).
All of them describe some kind of scalar-vector operation, and most of them constitute a scaling function for the set V : = R2.
In the symbolical expressions
u  =  æ
ç
è
u1
u2
ö
÷
ø		 ,    v  =  æ
ç
è
v1
v2
ö
÷
ø		   and  ||a||  is the length of a vector a,
and the arithmetical operations to the right from the  = - signs are normal number or vector operations.

II Problem 3. Which pairs of the following Verbal-Graphical representations describe the same operation in the plane R2?

Beside each symbolical expression you find a pull-down menu, with which you can choose a correct Figure A, B or C, or "none".
Maximum 8 x 1 = 8 points.

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Figure A
Figure B
Figure C

  Verbal Expression Corresponding Figure   Verbal Expression Corresponding Figure
1)
We add  c times  æ
ç
è		 1
1		 ö
÷
ø		  to the vector.
2) We add  c  times the unit vector to the vector.
3) The vector is reflected through x1-axis.  4) The result is one and a half times the vector.
5) The ordinary scalar multiplication of vectors.  6) The result is half of the ordinary scalar multiplication.
7) The sign of the second coordinate is changed.  8) The result is the vector itself.

II Problem 4. Which pairs of the following Symbolical-Graphical representations describe the same operation in the plane R2 ?

Beside each symbolical expression you find a pull-down menu, with which you can choose a correct Figure D, E or F, or "none".
Maximum 8 x 1 = 8 points.

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Figure D
Figure E
Figure F
  Symbolical Expression Corresponding Figure   Symbolical Expression Corresponding Figure
a) c*u  :=  3/2 u b)
c*u   :=   æ
ç
è		 1 + c u1
1 + c u2		 ö
÷
ø
c) c*u  :=  c u d)
c*u  := æ
ç
è		 - u1
 u2		 ö
÷
ø
e)
c*u  :=  u +  æ
ç
è		 c
c		 ö
÷
ø
f)
c*u  :=   c-u
g) c*u  :=   3c/2 u h) c*u  :=  c/2 u

III Scaling Functions: VSG Production Tasks
The following sketches (example + 4 tasks) contain graphical representations of some scaling functions.

They are some kind of dynamical "Puzzles", for which you are requested to produce verbal and symbolical forms.

Each produced expression is worth 3 points.
Maximum (2 x 4) x 3 = 24 points.


About the Sketches, Puzzles and Hints

The Sketches contain Hint Buttons, which will show you guiding objects like points and other useful geometrical elements used in constructions.

We ask that you tell how much you used Hints and how much you think they helped! For this, use the multiple choice questions after each Puzzle text input area.

To clear the extra stuff just push Hide hints and Show hints again.

An active Sketch can be reset reset by pushing 'R' (restart).

The idea of the Puzzles should be made clear through the following Example Puzzle.

III EXAMPLE PUZZLE

Problems
Maximum 0 points.
III Problem a. Express the operation verbally.
Maximum 0 points. 		III Problem b. Express the operation in symbolical form.
Maximum 0 points.
This page requires a Java capable browser.
Hints
Feel free to use the hints (Hint Buttons inside the Sketch).
But anyway, tell after answering each Puzzle problems, using the multiple choice buttons, whether you used Hints and how much they helped you.

Hint 1: Two construction points are shown.

Hint 2: The geometrical construction.

In the real Puzzles you should write your answers to the answer areas below the sketch.

Verbal example answer to the Example Puzzle:

Symbolical example answer to the Example Puzzle:

III PUZZLE 1

Problems (total 6 points)
III Problem 5. Express the operation verbally.
Maximum 3 points. 		III Problem 6. Express the operation in symbolical form.
Maximum 3 points.
This page requires a Java capable browser.
Hints
Feel free to use the hints (Hint Buttons inside the Sketch).
But anyway, tell after answering each Puzzle problems, using the multiple choice buttons, whether you used Hints and how much they helped you.

Hint 1: Two construction points are shown.

Hint 2: The geometrical construction.

Please write your answers to the answer areas below the sketch.

Your answer to III Puzzle 1, Problem 5 (verbal expression):

Your answer to III Puzzle 1, Problem 6 (symbolical expression):

Did you use Hints? not at all some all
If you used Hints, how much did they help you? not at all some crucial

III PUZZLE 2

Problems (total 6 points)
III Problem 7. Express the operation verbally.
Maximum 3 points. 		III Problem 8. Express the operation in symbolical form.
Maximum 3 points.
This page requires a Java capable browser.
Hints
Feel free to use the hints (Hint Buttons inside the Sketch).
But anyway, tell after answering each Puzzle problems, using the multiple choice buttons, whether you used Hints and how much they helped you.

Hint 1: One construction point is shown.

Hint 2: Adds two lines.

Hint 3: The geometrical construction.

Please write your answers to the answer areas below the sketch.

Your answer to III Puzzle 2, Problem 7 (verbal expression):

Your answer to III Puzzle 2, Problem 8 (symbolical expression):

Did you use Hints? not at all some all
If you used Hints, how much did they help you? not at all some crucial

III PUZZLE 3

Problems (total 6 points)
III Problem 9. Express the operation verbally.
Maximum 3 points. 		III Problem 10. Express the operation in symbolical form.
Maximum 3 points.
This page requires a Java capable browser.
Hints
Feel free to use the hints (Hint Buttons inside the Sketch).
But anyway, tell after answering each Puzzle problems, using the multiple choice buttons, whether you used Hints and how much they helped you.

Hint 1: Two construction points are shown.

Hint 2: More points and segments.

Hint 3: The geometrical construction.

Please write your answers to the answer areas below the sketch.

Your answer to III Puzzle 3, Problem 9 (verbal expression):

Your answer to III Puzzle 3, Problem 10 (symbolical expression):

Did you use Hints? not at all some all
If you used Hints, how much did they help you? not at all some crucial

III PUZZLE 4

Problems (total 6 points)
III Problem 11. Express the operation verbally.
Maximum 3 points. 		III Problem 12. Express the operation in symbolical form.
Maximum 3 points.
This page requires a Java capable browser.
Hints
Feel free to use the hints (Hint Buttons inside the Sketch).
But anyway, tell after answering each Puzzle problems, using the multiple choice buttons, whether you used Hints and how much they helped you.

Hint 1: Two construction points are shown.

Hint 2: Further Hint.

Please write your answers to the answer areas below the sketch.

Your answer to III Puzzle 4, Problem 11 (verbal expression):

Your answer to III Puzzle 4, Problem 12 (symbolical expression):

Did you use Hints? not at all some all
If you used Hints, how much did they help you? not at all some crucial

IV Student Feedback

Problem 13.
Extra bonus 2 points.
Here we ask you to tell what do you think about this kind of Problems and working on this kind of worksheet.



Anyone interested in this kind of worksheets,
please contact
Martti.Pesonen@Joensuu.Fi