MATH 1503

Justine Gauthier, Sep 3, 2019

Table of Contents

  1. Chapter 1
    1. Linear Equations in 2 Variables
    2. System of Equations
    3. 2D Vectors
    4. Conics
  2. Chapter 2
    1. Linear Transformations
    2. Visualizing Linear Transformations
    3. Rotations: matrix
    4. Eigenvectors and Eigenvalues
    5. Dot Product Insight
  3. Chapter 4
    1. Vector equation of a line (2D)
  4. Recreational Maths
    1. Proof Without Words
    2. Tower of Hanoi
    3. Triangle Paradox (Missing square puzzle)

1.

Chapter 1

  • 1. Linear Equations in 2 Variables
  • 2. System of Equations
  • 3. 2D Vectors
  • 4. Conics

1.1.

Linear Equations in 2 Variables

– Justine Gauthier

1.2.

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1.3.

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1.4.

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2.

Chapter 2

  • 1. Linear Transformations
  • 2. Visualizing Linear Transformations
  • 3. Rotations: matrix
  • 4. Eigenvectors and Eigenvalues
  • 5. Dot Product Insight

2.1.

Linear Transformations

This applet shows the effect of a linear transformation [math]T:\mathbb{R}^2 \rightarrow \mathbb{R}^2[/math]. The effects of [math]T[/math] on the blue vector [color=#0000FF][math]\vec{v}[/math][/color] and the [color=#0000FF]blue triangle[/color] are depicted as the red vector [color=#FF0000][math]T(\vec{v})[/math][/color] and the [color=#FF0000]red triangle[/color]. The matrix corresponding to [math]T[/math] is called [math]A[/math]. Recall that the column vectors of [math]A[/math] are given by [math]T(\vec{e}_1)[/math] and [math]T(\vec{e}_2)[/math], where [math]\vec{e}_1=\begin{pmatrix}1\\0\end{pmatrix}[/math] and [math]\vec{e}_2=\begin{pmatrix}0\\1\end{pmatrix}[/math]. Manipulate [math]T(\vec{e}_1)[/math] and [math]T(\vec{e}_2)[/math] to define a new linear transformation and see its corresponding matrix. You may also manipluate [color=#0000FF][math]\vec{v}[/math][/color] and the [color=#0000FF]blue triangle[/color].

[b]Exercises[/b] For the following four exercises, find the matrix for the linear transformation corresponding to [list=5] [*] scaling by the factor 1/2. [*] reflection across the line [math]y=-x[/math]. [*] 180 degree rotation about the origin. [*] projection onto the [math]y[/math]-axis. [/list] For the next four exercises, describe the linear transformation given by the matrix [list=5] [*] [math]\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}[/math] [*] [math]\begin{bmatrix}0.5 \quad 0.5\\0.5 \quad 0.5\end{bmatrix}[/math] [*] [math]\begin{bmatrix}0 \quad 1\\1 \quad 0\end{bmatrix}[/math] [*] [math]\begin{bmatrix}0 \quad 2\\2 \quad 0\end{bmatrix}[/math] [/list]
– abulawa

2.2.

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2.3.

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2.4.

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2.5.

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3.

Chapter 4

  • 1. Vector equation of a line (2D)

3.1.

Vector equation of a line (2D)

Click and drag the points A and D to define the line.[br]The direction of the line is controlled by the direction vector [b]d[/b] (using point D).[br]The line passes through the point A in a direction defined by the vector [math]\lambda \mathbf{d}[/math] where [math]\lambda[/math] is a parameter which can be varied with the slider.[br]The vector equation of the line is a parametric equation of the form [math]\mathbf{r}=\mathbf{a}+\lambda \mathbf{d}[/math].
– Dr Adrian Jannetta

4.

Recreational Maths

  • 1. Proof Without Words
  • 2. Tower of Hanoi
  • 3. Triangle Paradox (Missing square puzzle)

4.1.

Proof Without Words

Drag the points in the sketch below.[br]What do you notice? What does this prove? How does this prove it?
– Steve Phelps

4.2.

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4.3.

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