Linear Algebra

Ana D Todorova, Apr 8, 2016

This book contains materials for upper high school Linear algebra and analytic geometry. It has three chapters: 1. Vectors, 2. Dot Product of Vectors and 3. Determinants. Offered materials in each of the chapters support three modes of description: geometric, algebraic and abstract (Hillel, 2000). They are in close connection to three modes of thinking: synthetic-geometric, arithmetic and analytic-structural (Sierpinska, 2000). The materials in the GeoGebra Book aim to help students in recognizing, utilizing and translating between the multiple modes of thinking of the corresponding concept. In particular, they may help in understanding the bi-linearity properties of the dot product and the multi-linearity property of determinants. Additional Reading: http://www.math.hu-berlin.de/~todorova/

Table of Contents

  1. Vectors
    1. Vector Addition, Commutative Property
    2. Associative Property of Vector Addition
  2. Dot Product of Vectors
    1. Dot Product of Vectors
    2. Additive Property of the Dot Product
  3. Determinants
    1. Scaling Property of Determinants
    2. Additive Property of Determinants

1.

Vectors

The applets in this book aim to support the understanding of the commutative and associative properties of vector addition. For additional reading see: Filler, A.; Donevska-Todorova, A. (2012). Der Vektorbegriff. Verschiedene Wege zu seiner Einführung. In: [i]Mathematik Lehren 172[/i](6), S. 47-51.

  • 1. Vector Addition, Commutative Property
  • 2. Associative Property of Vector Addition

1.1.

Vector Addition, Commutative Property
– Ana D Todorova

1.2.

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2.

Dot Product of Vectors

The dot product is defined by three axioms. The applets in this book aim to support the understanding of these axioms. For additional reading see: Donevska-Todorova, A. (2015). Conceptual Understanding of Dot Product of Vectors in a Dynamic Geometry Environment. [i]The Electronic Journal of Mathematics & Technology 9[/i](3).

  • 1. Dot Product of Vectors
  • 2. Additive Property of the Dot Product

2.1.

Dot Product of Vectors

Dot Product of Vectors
Dot Product of Vectors
What is the relation between the area of a quadrilateral and the dot product of two vectors [color=#1551b5]u[/color] and [color=#c51414]v[/color]?
– Ana D Todorova

2.2.

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3.

Determinants

Determinants are defined with three axioms. The applets in this book aim to help in understanding these axioms. For additional reading see: Donevska-Todorova, A. (2016). Thinking Modes, with or without technology? [i]Proceedings ICME 2016[/i] (accepted for publishing).

  • 1. Scaling Property of Determinants
  • 2. Additive Property of Determinants

3.1.

Scaling Property of Determinants

Visualization of the Axiom 3a
1. Double the length of one side of the parallelogram [i]ABCD[/i]. (Set one of the sliders at 2 and the other one at 1). [br] a) How does it affect its area? [br] b) Write your answer with determinants’ notation. [br] c) Compare the length and the direction of the vectors [math]u[/math] and [math]eu[/math] (or [math]v[/math] and [math]kv[/math] ). What is their relation to the determinant?[br][br]2. Double the lengths of both sides of the parallelogram . How does it affect [br] a) its area?[br] b) the entries in each row of the determinant?[br] c) the value of the determinant? [br][br]3. Double one of the sides of the parallelogram and triple the other one. [br] a) How does it affect its area? Write your answer with determinants’ notation.[br] b) Compare the result to the previous exercise.[br] c) Explore for other real numbers and generalize your answer.[br][br]4. Set both sliders at 1, B at (1,0) and D at (0,1). [br] a) Which geometric figure is obtained? [br] b) Which of the axioms for determinants is provided?[br][br]5. Can a determinant represent area of a rectangle? Investigate how!
– Ana D Todorova

3.2.

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