Linear Algebra

Dynamical Problem Sheet 1: Binary Operations

- towards the concept vector binary operation in abstract vector spaces
DEAR STUDENT
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Purpose of this Worksheet
The purpose of this Worksheet is to reinforce the understanding of abstract binary operations, especially mappings
(vector,vector) -> vector
where the vectors are elements in a certain set, whose elements are called vectors.

Contents

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

IV Student Feedback

Brief Instructions

This example sketch visualizes the standard vector addition +, multiplication by scalars and linear combinations of plane vectors.
This page requires a Java capable browser. In these JavaSketchpad applet constructions which we shall call Sketches, you can usually drag (move in the usual way using the mouse) the red points. These usually represent either points in the plane or numbers on a line.
The red point labeled u in the Sketch beside stands for the endpoint of the vector u 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 u o v is the image of the pair (u,v) with respect to the binary operation o in question; we call it the result of the operation.

A Sketch is activated by a mouse click. An active Sketch can be reset by pressing the key 'R' (restart).

I Binary Operations: Definition Identification Tasks

In each of the following 6 plane geometry figures (dynamical Sketches) you see two vectors u and v and a resulting operation u o v.
You are supposed to make clear which of them are internal binary operations in the plane.

Let us recall the definition of the concept "internal binary operation in a set" (in Finnish: laskutoimitus jossakin joukossa):


DEFINITION OF INTERNAL BINARY OPERATION
Let V be a nonempty set. Any two variable function o: V×V®V is called a binary operation in V.
This means: for every ordered pair (u,v) in V×V there must be exactly one image element in V, called the result, and denoted by u o v.

I Problem 1. In which of the Figures A-F the operation is a binary operation in 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 binary operation? yes no I don't know
Is Figure B a binary operation? yes no I don't know
Is Figure C a binary operation? yes no I don't know
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Figure D
Figure E
Figure F
Is Figure D a binary operation? yes no I don't know
Is Figure E a binary operation? yes no I don't know
Is Figure F a binary operation? yes no I don't know
I Problem 2. In the case(s) where the operation is not a binary operation in R2, explain why not.
Maximum 4 points.

Your answer to I Problem 2:

II Binary Operations: 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 vector-vector operation, and most of them constitute an internal binary operation (in Finnish: sisäinen laskutoimitus) in 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 whole plane R2?

Beside each verbal 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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This page requires a Java capable browser.
Figure A
Figure B
Figure C
  Verbal Expression Corresponding Figure   Verbal Expression Corresponding Figure
1) The ordinary addition of vectors. 2) Twice the vector connecting the first and second vector.
3) The mean value of the vectors rotated by 90 degrees counterclockwise. 4) The result is the mean value of all the coordinates.
5) Difference between first and half the second.  6) The result is half of the first vector.
7) To the first vector we add a vector parallel to the positive x1-axis, with the same length as the second vector. 8) The result is the zero vector.

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 A, B or C, or "none".
Maximum 8 x 1 = 8 points.

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This page requires a Java capable browser.
This page requires a Java capable browser.
Figure A
Figure B
Figure C
  Symbolical Expression Corresponding Figure   Symbolical Expression Corresponding Figure
a) u o v := 1/2 u + 1/2 v b) u o v := 2(v -u)
c) u o v := v + u d)
u o v :=  æ
ç
è		 u1
u2		 ö
÷
ø		  +  æ
ç
è		 v1
0		 ö
÷
ø
e)
u o v :=  æ
ç
è		 ||v||
0 		 ö
÷
ø		  + u
f)
u o v :=  æ
ç
è		 u1
v2		 ö
÷
ø		  +  æ
ç
è		 v1
u2		 ö
÷
ø
g)
u o v :=  1
2		 (v-u) + u
h) u o v := u1v1 + u2v2

III Binary Operations: VSG Production Tasks
The following Sketches (Example + 4 Tasks) contain graphical representations of some internal binary operations.

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

Each expressions is worth 3 point.
Maximum (2 x 4) x 3 = 24 points.


About Sketches, Puzzles and Hints

The Sketches contain Hint Buttons, that show you helpful objects like points or other geometrical elements used in the construction.

We ask you to tell us, after answering a problem, whether you used hints, and how much did they help you. For that, there are multiple choice questions after each Puzzle.

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

An active Sketch can be reset by pressing the key 'R'.

The idea of the Puzzles should become clear by the following Example.

III EXAMPLE PUZZLE

Problems
Maximum 0 points.
III Problem a. Express the operation verbally.
Maximum 0 points. 		III Problem b. Express the binary operation in symbolical form.
Maximum 0 points.
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Hints
Feel free to use the Hints in the Sketch.

Hint 1: The position of the vectors is interchanged.

Hint 2: Shows a line.

In the Puzzles you are supposed to write your answers to the answer areas below the sketch, see Example Answers below.

Tell using the multiple choice questions whether you used Hints and how much they helped you.

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 binary operation in symbolical form.
Maximum 3 points.
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Hints
Feel free to use the Hints in the Sketch, but please, tell us about it by using the appropriate multiple choice questions below.

Hint 1: Shows two construction points.

Hint 2: Shows the 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 binary operation in symbolical form.
Maximum 3 points.
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Hints
Feel free to use the Hints in the Sketch, but please, tell us about it by using the appropriate multiple choice questions below.

Hint 1: Two helpful points are shown.

Hint 2: Three segments are added.

Hint 3: Shows the 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 binary operation in symbolical form.
Maximum 3 points.
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Hints
Feel free to use the Hints in the Sketch, but please, tell us about it by using the appropriate multiple choice questions below.

Hint 1: Two helpful points are shown.

Hint 2: A line and a segment are added.

Hint 3: Shows the 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 binary operation in symbolical form.
Maximum 3 points.
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Hints
Feel free to use the Hints in the Sketch, but please, tell us about it by using the appropriate multiple choice questions below.

Hint 1: Two construction points are shown.

Hint 2: A circle.

Hint 3: Another circle.

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. Here we ask you to tell what do you think about this kind of Problems and working on this kind of worksheet.



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please contact
Martti.Pesonen@Joensuu.Fi