Linear Algebra

Dynamical worksheet 4: Linear Transforms in the Plane

- projection, reflection, dilation ja rotation from the geometrical point of view
- properties of linearity
- matrix representation
- eigenvalues and eigenvectors (in another document LinearTransform2.htm)
DEAR STUDENT
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Purpose of this Worksheet
The purpose of this Worksheet is to reinforce the understanding of
basic linear functions in the euclidean plane
the linearity properties
matrix representation of linear a transform
eigenvalues and eigenvectors (in another document LinearTransform2.htm)

Contents

Short instructions
0 Basic Linear Functions
I Identification and Puzzles
II Student Feedback

Short 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 coloured 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'.

O Basic Linear Functions

The following Javasketchpad sketches show the elementary linear functions
projection, reflection, dilation, and rotation,
in the Euclidean Plane R2.

The arithmetic operations in the plane are, of course, the ordinary addition of vectors coordinate-wise and the multiplication by scalars.
With the computer mouse you can drag the variable u and control the transformations in suitable ways. Feel free to explore!

O A Projection on horizontal axis. (maximum 7 points)

This page requires a Java capable browser. 		Problem 0 A. Let us investigate the projection to the horizontal axis as a function R2 ® R2.

1. The image of the point (2 4)T is (4 0)T (2 0)T 2 (0 2)T don't know

2. The pre-image of the point (2 4)T is
(4 2)T (2 0)T 2 empty don't know

3. The pre-image of the point (2 0)T is
(4 2)T (2 4)T (2 0)T {(2 y)T | y in R} don't know

You may use animations in the following:
4. What is the image of the segment (-6 2)T ® (-4 -2)T ?
(-6 -4)T segment (-6 0)T ® (-4 0)T {(-6 0)T,(-4 0)T} don't know
5. What is the image of a circle? point circle segment don't know

6. Is the function L an injection? yes no don't know
7. Is the function L a surjection? yes no don't know

O B Reflection over the horizontal axis (maximum 5 points)

This page requires a Java capable browser. 		Problem O B. Let us investigate the reflection over the horizontal axis as a function R2 ® R2.

2. The image of the point (2 4)T is
(4 0)T (2 -4)T 2 (-2 4)T don't know

2. The pre-image of the point (2 4)T is
(4 2)T (-2 -4)T 2 (2 -4)T don't know

3. What is the image of a circle? point circle segment don't know

4. Is the function L a bijection? yes no don't know

5. For reflection L, what is (LoL)(u) =

0 C Dilations in the Plane (maximum 4 points)

This page requires a Java capable browser. 		Problem 0 C. Let us investigate the dilation as a function R2 ® R2:
C æ
è 		x1
x2
 ö
ø 		: = æ
è 		cx1
 x2
 ö
ø 		       D æ
è 		x1
x2
 ö
ø 		: = æ
è 		 x1
dx2
 ö
ø
To begin with, use the left hand side options in the Sketch.

1. What is the image of a segment under the function C, when c is not zero?
segment circle triangle line don't know
2. What is the image of a circle under the function C, when c is not zero?
segment ellipse parabola line don't know

3. Is the function C a bijection, when c < 0 ? yes no don't know
Let us now investigate a composite function (composition), where coordinates are scaled by constants that can be varied using the "Change"-options in the bottom.
(D °C) æ
è 		x1
x2
 ö
ø 		= æ
è 		cx1
dx2
 ö
ø
Use now the options for compositions in the bottom (Compose C and D) and for example a circular animation.
4. For what kind of values c and d the image of an origin-centered circle is a circle, under the composed mapping DoC?
It must be:

O D Rotation and Dilation in the Plane (maximum 4 points)

This page requires a Java capable browser. 		Problem O D. Let us investigate rotation compositions of dilation ja rotation as functions R2 ® R2:

To begin with, use the left hand side options in the Sketch.

1. What is the image of a segment under the function R?
segment circle triangle line don't know
2. What is the image of a circle under the function R?
segment ellipse circle line don't know

3. Find an angle of rotation from the interval [0, 2p] so that the image of the point (4 1)T is in the positive vertical axis.
The angle is:

Now take also the Buttons on the right hand side.
4. Is the function D a bijection for all real numbers d ? yes no don't know

Let us now investigate the composed function DoR (Compose R and D). You can vary these with the "Change"-options at the bottom.
5. Reset (keyboard "R") the figureand choose "Compose R and D". Find the matrix of the composed function (approximately!):

6. How does the function DoR differ from the function RoD (when we only look at the final result) ?
not at all much don't know

I Linear Transforms: Identification and Puzzles

In each of the following Sketches you see a construction showing the behaviour of a function L.
You should solve the problems beside the Sketch. In them you must either choose one alternative, or give numerical or verbal inputs.

Let us remind the definition of a linear function:

DEFINITION OF LINEAR FUNCTION (transformation)
Let (V, +, ) and (W, +, ) be linear spaces (vector spaces) over the same field (here the real scalar field) and L: V®W a function.
The function L is linear, if the following conditions hold:
(i) L(u + v) = L(u) + L(v) for all u, v in the set V
(ii) L(c u) = c L(u) for all u in the set V, for all scalars c.

Short instructions and a little training

This Example Sketch shows a plane function F. The image of the variable vector u is F(u), and it (possibly) moves, when you drag the variable u with the mouse.

Maximum 2 points.

You can work freely by playing with the buttons and moving the objects. However, the success often depends on following the instructions
The Sketch is activated by a click of the mouse. An active Sketch is reset by pushing the keyboard letter 'R'.
Left side: addition Right side: scaling
Images: IMAGES ON F

This page requires a Java capable browser. 		Training
1. What is the image of the vector (3 4)T?
()T
2. What is the vector whose image is (3 4)T?
()T

The following problems are common to the Sketches in this part:

1. Is the function F linear?
2. What geometric construction forms the function F (give a verbal expression) ?
3. Form a symbolical expression for F.
4. If F is linear, express its matrix (approximation).

Exemple solutions
1. The function is not linear. This can be seen in many ways:
- the conditions (i) or (ii) of the definition.
- the lacking of some property that linear functions have.
Here it would be simpliest to notice that the zero vector is not mapped to zero in this function (a proved property!)
2. You can easily find the translation vector a, with which F(u) is got from the vector u, but how?
3. F(u) = u + a = u + (2 1)^T.

I Puzzle 1

Maximum 4 points.
This page requires a Java capable browser.

I Problems to Puzzle 1

1. Is the function L linear? yes no don't know

2. What geometric construction forms the function L (give a verbal expression) ?

3. Form a symbolical expression for L:

4. If L is linear, express its matrix (approximation):


Your comments to the Problem I Puzzle 1:

I Puzzle 2

Maximum 4 points.
This page requires a Java capable browser.

I Problems to Puzzle 2

1. Is the function L linear? yes no don't know

2. What geometric construction forms the function L (give a verbal expression) ?

3. Form a symbolical expression for L:

4. If L is linear, express its matrix (approximation):


Your comments to the Problem I Puzzle 2:

I Puzzle 3

Maximum 4 points.
This page requires a Java capable browser.

I Problems to Puzzle 3

1. Is the function L linear? yes no don't know

2. What geometric construction forms the function L ? (give a verbal expression)

3. Form a symbolical expression for L:

4. If L is linear, express its matrix (approximation):


Your comments to the Problem I Puzzle 3:

I Puzzle 4

Maximum 4 points.
This page requires a Java capable browser.

I Problems to Puzzle 4

1. Is the function L linear? yes no don't know

2. What geometric construction forms the function L ?

3. If L is linear, express its matrix (approximation):

4. What hints did you use?


Your comments to the Problem I Puzzle 4:

I Puzzle 5

Maximum 4 points.
This page requires a Java capable browser.

I Problems to Puzzle 5

1. Is the function L linear? yes no don't know

2. What geometric construction forms the function L ?


Your comments to the Problem I Puzzle 5:

II Student Feedback

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