Linear Algebra

Dynamical worksheet 4b: Linear Functions in the Plane II

- properties of linearity
- matrix representation
- eigenvalues and eigenvectors
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Purpose of this Worksheet
The purpose of this Worksheet is to reinforce the understanding of
- matrix representation of linear mapping
- eigenvalues and eigenvectors of matrix and linear function

Contents

Short instructions
II Matrix representation
III Eigenvalues and eigenvectors

IV 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'.

II Linear Functions: Matrix Representation

In this part we consider transformation of vectors and different figures under linear functions in the plane V : = R2. Especially we investigate the effect of the matrix to the transformations.

Instructions

II Sketch 1: only the draggable domain vector u and its image L(u) are shown.

II Sketch 2: you may, in addition, use some pre-defined animations. You may find it convenient to look at each particular item separately - use the appropriate Hide buttons.

II Sketch 3: you also find the matrix A = AL of the function together with its determinant det(A). The matrix elements can be controlled using the sliding points a, b, c, d on the real axis in the lower part of the sketch.

II Sketch 4: finally you can use the matrix A = AL, its determinant det(A), trace tr(A) and the two eigenvalues eig(A). The matrix elements can again be controlled using the sliding points a, b, c, d on the real axis.

THE MATRIX REPRESENTATION OF LINEAR FUNCTION IN THE PLANE
When  2x2 matrix A of L]and  plane vector u,
the function   linear function L from plane to itselfis defined by
L(u)  =  image of u under L:=   =  matrix product=  A u.
Perhaps you still remember how the matrix of a linear function is obtained by calculations?
Namely: the images of the standard basis are the columns of the matrix!
Recall that the determinant of the 2x2-matrix A is
det(A) = ad - bc,
and the trace is the sum of the diagonal elements
tr(A) = a + d.
A scalar c is an eigenvalue of a linear function L (and its matrix) if
L(u) = cu.
for some nonzero vector u.

A non-zero vector u satistying the condition is called an eigenvector (for the eigenvalue c).

II Problems 1. In this Sketch only the draggable domain vector u and its image L(u) are shown. You may show/hide segments and the tracing facility Trace.
You may use the segment PQ as a tool.
Maximum 6 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
This page requires a Java capable browser. Using for example tracing:
a) How does the function L map straight lines, i.e. what is the image of a line like?

b) What is the image of a square like?

c) When you turn u around the origin counterclockwise, how does the image L(u) behave?
Using tracing or the segment tool PQ:
d) Which points are mapped to the vertical axis?

e) Which vectors u are parallel to L(u)?

f) Can you find fixed points, i.e. points u for which u = L(u) ?
What points?

Your comments to II Problems 1:

II Problems 2. Also in this Sketch the draggable domain vector u and its image L(u) are shown and, if needed, the coordinates. In addition, you can use some pre-defined animations. You can move the circle and segment of animation as you please.
It may be most advantageous to look at each item separately - use the Hide buttons where appropriate.
Maximum 8 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner. Again you may show/hide segments and use the segment PQ as a tool.
This page requires a Java capable browser. Using tracing or animations (then see Locus):
a) What geometrical figure is the image of a circle under the mapping L ?

b) What is the image of a segment under the mapping L ?

c) What are the images of the standard base vectors (1 0)T (0 1)T under the mapping L (see coordinates) ?
L(1 0)T =
L(0 1)T =
d) What is the matrix of the function L ?

e) What vectors u are parallel with their image L(u) ?

f) What are the fixed points, (i.e. points u for which u = L(u)) ?

Your comments to II Problems 2:

II Problems 3. In this Sketch you see the variable vector u and its image L(u), and in addition to the animations and coordinates the matrix A = AL and its determinant det(A).
Here you may change the matrix by dragging its elements, the points a, b, c, d on the real line. This means that you are allowed to change the linear function L.
Maximum 8 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
You can get all kinds of linear functions by controlling the elements!
This page requires a Java capable browser. a) What geometrical objects can the images of circles be under linear functions?

b) What geometrical objects can the images of segments be under linear functions?

c) What is the value of the determinant, when the image of a circle is a segment?
det(A) =
d) Move the animation circle to contain the origin and make the vector u turn around circle (use animation!).
What is the difference between the determinant value in cases, where the the image L(u) turns around the origin to the same vs. opposite direction when compared to the movement of u ?

e) Reset the Sketch. Without changing the matrix A, find by using the Sketch the eigenvalues and the eigenvectors of the function (and the matrix, of course):
eigval1(L) = , eigvect1(L) =
eigval2(L) = , eigvect2(L) =
f) Is the following conjecture correct:
If the determinant is positive, there are no real eigenvalues.
Prove the statement if it is correct, or find a counterexample.

Your comments to II Problems 3:

II Problems 4. In this Sketch you see, in addition to the vectors, animations, matrix and determinant, the trace tr(A) and eigenvalues eig(A).
Here you may change the matrix by dragging its elements a, b, c, d on the real line. This means that you are allowed to change the linear function L.
Maximum 8 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
You can get all kinds of linear functions by controlling the elements.
This page requires a Java capable browser. a) What is the effect of the individual movement of number b ?

b) Reset, show the representation of L and start the circle animation. Adjust the matrix elements so that the circle is mapped to a segment. What are the numbers:
det(A) =
eigval1(A) =
eigval2(A) =
c) What is the kernel of the function L ?
ker(L) =
d) Find a matrix A which maps a circle to a circle of the same size.

e) What is the determinant of your matrix ?
det(A) =
f) Find a simple relation involving the area of the interior of a circle, the area inside the image curve (locus) of that circle, and the determinant.

Your comments to II Problems 4:

III Linear Functions: Eigenvalues of Linear Functions

The following JSP-applet constructions contain illustration of the eigenvalues of linear mappings in the Euclidean plane R2, also in non-real situations.
Let us recall how the eigenvalues of a 2x2-matrix can be calculated:

CALCULATING EIGENVALUES
The eigenvalues of a 2x2-matrix
are the real or complex solutions of the characteristic equation
and with slight simplifications we get

III Problems 1: Eigenvalues as functions of the matrix elements
The same Sketch is also the complex plane, where the eigenvalues of the matrix are E1 and E2.
Maximum 5 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
This page requires a Java capable browser. Let us start by looking at the 2x2-matrix
and the roots of the characteristic equation
Let us keep a, c and d fixed and change the value of b by moving the corresponding white point.
a) For what value(s) b are the eigenvalues equal, i.e. it is a double root?
Find the answer experimentally on the screen:
b =
b) Make it sure by computation: (see. formula):

c) Explain the shape that appears on the screen when moving b through the real axis (tracing!).

Your comments to III Problems 1:

III Problems 2: Eigenvalues as functions of the matrix elements
The same Sketch is also the complex plane, where the eigenvalues of the matrix are E1 and E2.
Maximum 5 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
This page requires a Java capable browser. Let us again look at the 2x2-matrix
and the roots of the characteristic equation
Let us keep b, c and d fixed and change the value of a by moving the corresponding white point.
a) For what value(s) a are the eigenvalues equal, i.e. it is a double root?
Find the answer experimentally on the screen:
a =
b) Make it sure by computation: (see. formula):

c) Explain the shape that appears on the screen when moving a through the real axis (tracing!).

Your comments to III Problems 2:

III Problems 3: Eigenvalues as functions of the matrix elements
The same Sketch is also the complex plane, where the eigenvalues of the matrix are E1 and E2.
Maximum 5 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
This page requires a Java capable browser. Let us again look at the 2x2-matrix
and the roots of the characteristic equation

a) For what value pairs b, c are the eigenvalues real, independently from the choise of a and d?
Find the answer experimentally on the screen:

b) Make it sure by computation: (see. formula):

c) Change the matrix so that when you keep b,c fixed, then by moving a (or d) the eigenvalues make a circlular trace. For which knd of pairs a,d are the imaginary parts of the eigenvalues greatest (in absolute value)?

Your comments to III Problems 3:

III Problems 4: Eigenvalues of a linear function
The same Sketch is also the complex plane, where the eigenvalues of the matrix are E1 and E2.
Maximum 5 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
This page requires a Java capable browser.
a) Describe the effect of individual numbers a, b, c ja d to the image of a (fixed) vector u: b) Reset ('R') and keep a, c and d fixed. Use the segment animation and locus:
For which values of b it is possible to choose the segment of animations so, that the image is parallel with it? Try several values of b, for example b = 4, b = 0, b = -3. Can you then find a general rule (see. again formula): c) Choose to see segments O-u, O-L(u) and u-L(u), when you have a triangle O-u-L(u)-O. What can you say about the eigenvalues by looking at this triangle as the vector u is turned around the origin?

Your comments to III Problems 4:

III Problems 5: The rank of linear function
The same Sketch is also the complex plane, where the eigenvalues of the matrix are E1 and E2.
Maximum 8 points. 		The Sketch is activated by a click of the mouse.
An active Sketch is reset by keyboard 'R'.
Tracing is cleaned by the red cross in the lower right corner.
This page requires a Java capable browser.
a) Find some numbers a, b, c ja d, for which a circle is mapped to a segment. You can use combinations of circle animation, Locus and controls.
Then give the following numbers, corresponding to your choice above:
a = b =
c = d =
det(A) =
The rank of the matrix =

b) What was the dimension of the image space, the range?
dimL(R2) =
Dimension of the kernel?
dim(ker(L)) =

c) Reset all. Choose b = c = 3 and leave the animation circle unchanged. Find a matrix, whose linear function maps circles to circles.
a = d =

d) What can you say about the eigenvalues that are drawn in the same figure? Is that just a coincidence?


Your comments to III Problems 5:

II Student Feedback

Please tell us what do you think about these Exercises and working on this kind of worksheet in general.
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Updated 30.12.2012