Linear Algebra

4.9 Basic Matrix Transformations in \(R^{2}\) and \(R^{3}\)

Imron Rosyadi

Basic Matrix Transformations in \(R^{2}\) and \(R^{3}\)

Overview of Transformations

This lecture explores fundamental matrix transformations in \(R^{2}\) and \(R^{3}\).
These transformations have simple geometric interpretations.
They are crucial in fields like computer graphics, engineering, and physics.
We will cover reflections, projections, rotations, dilations, contractions, expansions, compressions, and shears.

Welcome to this session on basic matrix transformations. We’ll be looking at how matrices can geometrically transform vectors and shapes in 2D and 3D space. These concepts are foundational for many ECE applications. We’ll start with reflections and move through various types of transformations. Pay attention to the interactive examples, as they will help you visualize these abstract concepts.

Reflection Operators

Reflection operators map each point to its symmetric image about a fixed line or plane through the origin.
These are fundamental in geometry and computer graphics.

Note

Remember that these reflections are about lines or planes that contain the origin.
This ensures they are linear transformations.

Reflection operators are like looking into a mirror. Every point is mapped to its mirror image. For linear algebra, we specifically focus on reflections across lines or planes that pass through the origin. This ensures the transformation is linear and can be represented by a matrix.

Reflection Operators in \(R^{2}\)

Here are the standard matrices for reflections about coordinate axes and the line \(y=x\) in \(R^{2}\).

Reflection about the x-axis
\(T(x,y) = (x,-y)\)
Standard Matrix:
\[ \left[ \begin{array}{cc}1 & 0 \\ 0 & -1 \end{array} \right] \]

Reflection about the y-axis
\(T(x,y) = (-x,y)\)
Standard Matrix:
\[ \left[ \begin{array}{cc}-1 & 0 \\ 0 & 1 \end{array} \right] \]

Reflection about the line \(y=x\)
\(T(x,y) = (y,x)\)
Standard Matrix:
\[ \left[ \begin{array}{cc}0 & 1 \\ 1 & 0 \end{array} \right] \]

These matrices are derived by observing where the standard basis vectors, \((1,0)\) and \((0,1)\), are mapped. For example, reflecting \((1,0)\) about the x-axis gives \((1,0)\), and reflecting \((0,1)\) gives \((0,-1)\). These become the columns of the standard matrix. The same logic applies to the other reflections.

Interactive \(R^{2}\) Reflection

Visualize how a point is reflected across different lines in \(R^{2}\).
Adjust the coordinates of the point and observe its reflection.

viewof px_ref = Inputs.range([-5, 5], {step: 0.1, value: 2, label: "P_x"})
viewof py_ref = Inputs.range([-5, 5], {step: 0.1, value: 3, label: "P_y"})
viewof reflectionType_ref = Inputs.select(["x_axis", "y_axis", "y_equals_x"], {label: "Reflection Type", value: "x_axis"})

This interactive plot allows you to experiment with reflections. Change the original point’s coordinates using the sliders and select different reflection axes. Observe how the reflected point’s coordinates change according to the transformation matrices we just discussed. This hands-on experience should solidify your understanding of these basic transformations.

Reflection Operators in \(R^{3}\)

Reflections in \(R^{3}\) occur about coordinate planes.

Reflection about the \(xy\)-plane
\(T(x,y,z) = (x,y,-z)\)
Standard Matrix:
\[ \left[ \begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array} \right] \]

Reflection about the \(xz\)-plane
\(T(x,y,z) = (x,-y,z)\)
Standard Matrix:
\[ \left[ \begin{array}{ccc}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array} \right] \]

Reflection about the \(yz\)-plane
\(T(x,y,z) = (-x,y,z)\)
Standard Matrix:
\[ \left[ \begin{array}{ccc}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \]

Similar to \(R^{2}\), these matrices are formed by transforming the standard basis vectors \(\mathbf{e}_1=(1,0,0)\), \(\mathbf{e}_2=(0,1,0)\), and \(\mathbf{e}_3=(0,0,1)\). For instance, reflecting about the \(xy\)-plane means the \(x\) and \(y\) coordinates remain the same, while the \(z\) coordinate changes sign. So, \(\mathbf{e}_1 \to (1,0,0)\), \(\mathbf{e}_2 \to (0,1,0)\), and \(\mathbf{e}_3 \to (0,0,-1)\), forming the columns of the matrix.

Projection Operators

Projection operators map each point to its orthogonal projection onto a fixed line or plane through the origin. These are also called orthogonal projection operators.

Important

Orthogonal projection means the projected point is the closest point on the line/plane to the original point. The line segment connecting the original point and its projection is perpendicular to the line/plane.

Projection operators are like casting a shadow. If you shine a light perpendicular to a surface, the shadow cast is the orthogonal projection. In linear algebra, we project points onto lines or planes that pass through the origin. This is a crucial concept in signal processing and data analysis, for example, when reducing dimensionality.

Projection Operators in \(R^{2}\)

Standard matrices for orthogonal projections onto coordinate axes in \(R^{2}\).

Orthogonal projection onto the x-axis
\(T(x,y)=(x,0)\)
Standard Matrix:
\[ \left[ \begin{array}{cc}1 & 0 \\ 0 & 0 \end{array} \right] \]

Orthogonal projection onto the y-axis
\(T(x,y)=(0,y)\)
Standard Matrix:
\[ \left[ \begin{array}{cc}0 & 0 \\ 0 & 1 \end{array} \right] \]

For projection onto the x-axis, the y-component of any vector becomes zero, while the x-component remains unchanged. Thus, \((1,0) \to (1,0)\) and \((0,1) \to (0,0)\). This forms the columns of the projection matrix. Similarly for the y-axis.

Interactive \(R^{2}\) Projection

Visualize how a point is projected onto different lines in \(R^{2}\).
Adjust the point and observe its projection.

viewof px_proj = Inputs.range([-5, 5], {step: 0.1, value: 2, label: "P_x"})
viewof py_proj = Inputs.range([-5, 5], {step: 0.1, value: 3, label: "P_y"})
viewof projectionType_proj = Inputs.select(["x_axis", "y_axis", "y_equals_x"], {label: "Projection Type", value: "x_axis"})

Similar to reflections, this interactive tool helps visualize projections. You can adjust the original point and select different lines to project onto. Notice the dashed line connecting the original and projected points; this line is always perpendicular to the projection line, demonstrating the “orthogonal” aspect. The projection onto \(y=x\) is calculated as \(((x+y)/2, (x+y)/2)\).

Projection Operators in \(R^{3}\)

Standard matrices for orthogonal projections onto coordinate planes in \(R^{3}\).

Orthogonal projection onto the \(xy\)-plane
\(T(x,y,z) = (x,y,0)\)
Standard Matrix:
\[ \left[ \begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right] \]

Orthogonal projection onto the \(xz\)-plane
\(T(x,y,z) = (x,0,z)\)
Standard Matrix:
\[ \left[ \begin{array}{ccc}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right] \]

Orthogonal projection onto the \(yz\)-plane
\(T(x,y,z) = (0,y,z)\)
Standard Matrix:
\[ \left[ \begin{array}{ccc}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \]

For projections in \(R^{3}\), one of the coordinates becomes zero, effectively collapsing the point onto the chosen plane. For example, projecting onto the \(xy\)-plane means the \(z\)-coordinate becomes zero. So, \(\mathbf{e}_1 \to (1,0,0)\), \(\mathbf{e}_2 \to (0,1,0)\), and \(\mathbf{e}_3 \to (0,0,0)\), forming the columns of the matrix.

Rotation Operators

Rotation operators move points along arcs of circles centered at the origin.
We focus on counterclockwise rotations for positive angles \(\theta\).

Rotation in \(R^{2}\)

The images of the standard basis vectors \(\mathbf{e}_{1}=(1,0)\) and \(\mathbf{e}_{2}=(0,1)\) are:
\(T(\mathbf{e}_{1}) = (\cos \theta ,\sin \theta)\)
\(T(\mathbf{e}_{2}) = (-\sin \theta ,\cos \theta)\)

This gives the standard rotation matrix \(R_{\theta}\): \[ R_{\theta} = \left[ \begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right] \tag{1} \]

Rotation is a very common transformation, especially in robotics and computer graphics. The key idea here is that a rotation around the origin can be completely described by the angle of rotation. The standard basis vectors provide the columns for the rotation matrix. We define positive angles as counterclockwise rotations.

Visualizing \(R^{2}\) Rotation

Let’s visualize the transformation of basis vectors.

graph LR
    subgraph "Original Basis"
        e1_orig["e1 $$(1,0)$$"]
        e2_orig["e2 $$(0,1)$$"]
    end

    subgraph "Rotation Matrix"
        R_theta["$$R_{\theta}$$"]
    end

    subgraph "Rotated Basis"
        e1_rot["T(e1) $$(\cos\theta, \sin\theta)$$"]
        e2_rot["T(e2) $$(-\sin\theta, \cos\theta)$$"]
    end

    e1_orig --> R_theta
    e2_orig --> R_theta
    R_theta --> e1_rot
    R_theta --> e2_rot

    style e1_orig fill:#fff,stroke:#333,stroke-width:2px,color:#333
    style e2_orig fill:#fff,stroke:#333,stroke-width:2px,color:#333
    style e1_rot fill:#f9f,stroke:#333,stroke-width:2px,color:#333
    style e2_rot fill:#f9f,stroke:#333,stroke-width:2px,color:#333
    style R_theta fill:#ccf,stroke:#333,stroke-width:2px,color:#333

This Mermaid diagram illustrates how the standard basis vectors are transformed by the rotation matrix. The first column of \(R_{\theta}\) is \(T(\mathbf{e}_1)\), and the second column is \(T(\mathbf{e}_2)\). This visual representation helps connect the geometric interpretation with the algebraic matrix form.

Rotation Equations for \(R^{2}\)

If \(\mathbf{x} = (x,y)\) is a vector in \(R^{2}\), and \(\mathbf{w} = (w_{1},w_{2})\) is its image, then \(\mathbf{w} = R_{\theta}\mathbf{x}\) gives: \[ \begin{array}{l}w_{1} = x\cos \theta -y\sin \theta \\ w_{2} = x\sin \theta +y\cos \theta \end{array} \tag{2} \]

These are the rotation equations for \(R^{2}\).

Tip

For clockwise rotations, replace \(\theta\) with \(-\theta\) in the matrix. Since \(\cos(-\theta) = \cos\theta\) and \(\sin(-\theta) = -\sin\theta\), the matrix becomes: \[ R_{-\theta} = \left[ \begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array} \right] \]

These equations are simply the result of multiplying the rotation matrix by the column vector \([x, y]^T\). They show explicitly how the new coordinates \(w_1\) and \(w_2\) depend on the original coordinates \(x, y\) and the rotation angle \(\theta\). The tip about clockwise rotation is useful for practical applications.

Example 1: A Rotation Operator

Find the image of \(\mathbf{x} = (1,1)\) under a rotation of \(\pi /6\) radians (\(30^{\circ}\)) about the origin.

Here we apply the rotation matrix directly. The pyodide block allows you to execute this Python code right in the browser. You can modify the theta or x_vector values to see how the output changes. This demonstrates the practical application of the rotation matrix. The result matches the example in the provided text.

Interactive \(R^{2}\) Rotation

Rotate a shape (unit square) in \(R^{2}\) by varying the angle.

// Define input control for rotation angle
viewof angle_rot = Inputs.range([0, 360], {step: 1, value: 0, label: "Rotation Angle (degrees)"})

This interactive visualization shows a unit square rotating around the origin. Use the slider to change the rotation angle. The original square is blue, and the rotated square is red. This helps to intuitively grasp how the rotation matrix transforms entire objects, not just individual points.

Rotations in \(R^{3}\)

Rotations in \(R^{3}\) are described relative to an axis of rotation and a unit vector \(\mathbf{u}\) along that line. The right-hand rule determines the sign of the angle.

For coordinate axes, positive angles are counterclockwise when looking towards the origin along the positive axis.

In 3D, rotation is more complex than in 2D because you need to specify an axis of rotation in addition to the angle. The right-hand rule is a standard convention to define the positive direction of rotation. If you curl your fingers in the direction of rotation, your thumb points in the direction of the positive axis of rotation.

Standard Matrices for \(R^{3}\) Rotations

About the positive x-axis \[ \left[ \begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{array} \right] \]

About the positive y-axis \[ \left[ \begin{array}{ccc}\cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{array} \right] \]

About the positive z-axis \[ \left[ \begin{array}{ccc}\cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{array} \right] \]

Notice the patterns in these matrices. For rotation about the x-axis, the x-coordinate remains unchanged, and the 2x2 rotation matrix is applied to the y and z coordinates. Similarly for y and z axes. The y-axis rotation matrix is slightly different in the placement of the sine terms due to the standard basis vector ordering and right-hand rule convention.

Yaw, Pitch, and Roll

These angles describe the orientation of an aircraft or space shuttle relative to an \(xyz\)-coordinate system.

• Yaw: Rotation about the \(z\)-axis (side-to-side).
• Pitch: Rotation about the \(x\)-axis (nose up/down).
• Roll: Rotation about the \(y\)-axis (wing up/down).

A combination of these can be achieved by a single rotation about some axis through the origin.

This is a fantastic real-world application for ECE students. Understanding how these rotations are defined is critical in aerospace engineering, control systems, and even robotics. Space shuttles perform attitude adjustments by calculating a single rotation axis and angle to achieve the desired orientation, rather than performing three separate rotations. This highlights the importance of composite transformations.

General \(R^{3}\) Rotation

For completeness, the standard matrix for a counterclockwise rotation through an angle \(\theta\) about an axis determined by an arbitrary unit vector \(\mathbf{u} = (a, b, c)\) is:

\[ \left[ \begin{array}{llll}a^2 (1 - \cos \theta) + \cos \theta & ab(1 - \cos \theta) - c\sin \theta & ac(1 - \cos \theta) + b\sin \theta \\ ab(1 - \cos \theta) + c\sin \theta & b^2 (1 - \cos \theta) + \cos \theta & bc(1 - \cos \theta) - a\sin \theta \\ ac(1 - \cos \theta) - b\sin \theta & bc(1 - \cos \theta) + a\sin \theta & c^2 (1 - \cos \theta) + \cos \theta \end{array} \right] \tag{3} \]

Note

You can derive the axis-specific rotation matrices (Table 6) as special cases of this general formula by setting \(a,b,c\) to \((1,0,0)\), \((0,1,0)\), or \((0,0,1)\) respectively.

This general formula, while complex, is extremely powerful. It allows for rotation about any axis through the origin in 3D space. While we won’t derive it here, it’s good to be aware of its existence. The ability to derive the simpler coordinate-axis rotations from this general form provides a good check of understanding.

Dilations and Contractions

These operators scale the length of vectors. \(T(\mathbf{x}) = k\mathbf{x}\) for a nonnegative scalar \(k\).

• Contraction: If \(0 \leq k < 1\), the operator shrinks vectors.
• Dilation: If \(k > 1\), the operator stretches vectors.

Figure 4.9.3 (a) \(0 \leq k < 1\) (b) \(k > 1\)

If \(k=1\), it’s the identity operator.

Dilations and contractions are essentially scaling operations. They change the magnitude of vectors but not their direction. This is a very intuitive transformation. Think of zooming in or out on an image. The factor ‘k’ determines the amount of scaling.

Dilations and Contractions in \(R^{2}\)

Standard matrix for dilations/contractions in \(R^{2}\): \[ \left[ \begin{array}{cc}k & 0 \\ 0 & k \end{array} \right] \]

Dilations and Contractions in \(R^{2}\)

Interactive \(R^{2}\) Scaling

Scale a shape (unit square) in \(R^{2}\) by varying the factor \(k\).

// Define input control for scaling factor
viewof scaleFactor_scale = Inputs.range([0.1, 3], {step: 0.1, value: 1, label: "Scaling Factor (k)"})

The standard matrix for dilation/contraction is simple: a scalar multiple of the identity matrix. The interactive plot allows you to see this in action. When ‘k’ is less than 1, the square contracts; when ‘k’ is greater than 1, it dilates. This uniform scaling is what differentiates it from expansions and compressions.

Dilations and Contractions in \(R^{3}\)

Standard matrix for dilations/contractions in \(R^{3}\): \[ \left[ \begin{array}{ccc}k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{array} \right] \]

The principle is the same as in \(R^{2}\), just extended to three dimensions. Each coordinate is scaled by the factor \(k\). This type of transformation is used in 3D graphics for resizing objects uniformly.

Expansions and Compressions

These operators scale only one coordinate by a factor \(k\). - Compression: \(0 \leq k < 1\). - Expansion: \(k > 1\).

Compression/Expansion in x-direction \(T(x,y) = (kx,y)\) Standard Matrix: \[ \left[ \begin{array}{cc}k & 0 \\ 0 & 1 \end{array} \right] \]

Compression/Expansion in y-direction \(T(x,y) = (x,ky)\) Standard Matrix: \[ \left[ \begin{array}{cc}1 & 0 \\ 0 & k \end{array} \right] \]

Unlike dilations/contractions, expansions/compressions are non-uniform scaling operations. They only affect one dimension, either stretching or squishing the object along that specific axis. This is useful for reshaping objects in a non-proportional way.

Interactive \(R^{2}\) Expansion/Compression

Scale a shape (unit square) along a specific axis in \(R^{2}\).

viewof compExpFactor_ce = Inputs.range([0.1, 3], {step: 0.1, value: 1, label: "Scaling Factor (k)"})
viewof compExpDirection_ce = Inputs.select(["x", "y"], {label: "Direction", value: "x"})

Use this interactive tool to see how a square is compressed or expanded along either the x or y-axis. Notice that only one dimension changes, which results in a rectangle instead of a square (unless k=1). This is distinct from dilation, where both dimensions are scaled equally.

Shears

A shear transformation shifts points parallel to one axis, proportional to their distance from that axis.

Shear in the x-direction by factor \(k\)
\(T(x,y) = (x+ky,y)\)
Standard Matrix:
\[ \left[ \begin{array}{cc}1 & k \\ 0 & 1 \end{array} \right] \]

Shear in the y-direction by factor \(k\)
\(T(x,y) = (x,y+kx)\)
Standard Matrix:
\[ \left[ \begin{array}{cc}1 & 0 \\ k & 1 \end{array} \right] \]

Tip

A shear leaves points on the axis of the shear fixed. For an x-shear, points on the x-axis (\(y=0\)) remain unchanged. For a y-shear, points on the y-axis (\(x=0\)) remain unchanged.

Shears are often visualized as “tilting” or “slanted” transformations. Imagine pushing the top of a deck of cards sideways while keeping the bottom fixed. That’s an x-shear. The amount of shift ky is proportional to the y coordinate. This is commonly used in typography and graphical design.

Example 2: Effect of Matrix Operators on the Unit Square

Describe the matrix operator and show its effect on the unit square.

Matrix \(A_{1}\) \[ A_{1}={\left[\begin{array}{l l}{1}&{2}\\ {0}&{1}\end{array}\right]} \] This is a shear in the x-direction by a factor 2.

Matrix \(A_{2}\) \[ A_{2}={\left[\begin{array}{l l}{1}&{-2}\\ {0}&{1}\end{array}\right]} \] This is a shear in the x-direction by a factor -2.

Matrix \(A_{3}\) \[ A_{3}={\left[\begin{array}{l l}{2}&{0}\\ {0}&{2}\end{array}\right]} \] This is a dilation with factor 2.

Matrix \(A_{4}\) \[ A_{4}={\left[\begin{array}{l l}{2}&{0}\\ {0}&{1}\end{array}\right]} \] This is an expansion in the x-direction with factor 2.

This example directly applies the knowledge of standard matrices to identify the type of transformation. It’s a good exercise to mentally, or visually, confirm how each matrix would transform the unit square. For instance, for \(A_1\), the point \((1,1)\) becomes \((1+2\cdot1, 1) = (3,1)\).

Interactive \(R^{2}\) Shear

Apply a shear transformation to a unit square.

viewof shearFactor_shear = Inputs.range([-2, 2], {step: 0.1, value: 0, label: "Shear Factor (k)"})
viewof shearDirection_shear = Inputs.select(["x", "y"], {label: "Direction", value: "x"})

This interactive plot demonstrates x-direction and y-direction shears. Adjust the shear factor k and the direction. Observe how the square “tilts”. Note that the side of the square along the shear axis remains fixed, while the opposite side shifts. A positive k creates a shear in the positive direction, and a negative k creates a shear in the negative direction.

Orthogonal Projection onto a Line Through the Origin

The standard matrix \(P_{\theta}\) for orthogonal projection onto a line \(L\) through the origin, making an angle \(\theta\) with the positive \(x\)-axis, is:

\[ P_{\theta} = \left[ \begin{array}{cc}\cos^{2}\theta & \sin \theta \cos \theta \\ \sin \theta \cos \theta & \sin^{2}\theta \end{array} \right] = \left[ \begin{array}{cc}\cos^{2}\theta & \frac{1}{2}\sin 2\theta \\ \frac{1}{2}\sin 2\theta & \sin^{2}\theta \end{array} \right] \tag{4} \]

Figure 4.9.5

This is a more general projection than just onto the x or y axes. The angle \(\theta\) defines the line. This formula is derived by projecting the standard basis vectors onto the line \(y = (\tan \theta) x\). The trigonometric identities are often useful for simplifying the expressions.

Example 3: Orthogonal Projection onto a Line

Find the orthogonal projection of \(\mathbf{x} = (1,5)\) onto the line through the origin that makes an angle of \(\pi /6\) (\(30^{\circ}\)) with the positive \(x\)-axis.

This example demonstrates the application of the general projection formula. We calculate the projection matrix for \(\theta = \pi/6\) and then multiply it by the given vector \(\mathbf{x}\). The result shows the coordinates of the point \((1,5)\) projected onto the line at \(30^\circ\). This can be used in various signal processing contexts, such as projecting a signal onto a basis vector.

Reflections About Lines Through the Origin

The operator \(H_{\theta}:R^{2} \to R^{2}\) maps each point into its reflection about a line \(L\) through the origin that makes an angle \(\theta\) with the positive \(x\)-axis.

Figure 4.9.6

The standard matrix for \(H_{\theta}\) can be derived from the projection matrix \(P_{\theta}\): \(H_{\theta} = 2P_{\theta} - I\)

This leads to: \[ H_{\theta} = \left[ \begin{array}{cc}\cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{array} \right] \tag{6} \]

This is an interesting connection between projection and reflection. Geometrically, if you project a vector onto a line, and then extend the vector by the same amount on the other side of the line, you get the reflection. The formula \(H_{\theta} = 2P_{\theta} - I\) captures this relationship algebraically. The matrix for reflection about an arbitrary line is also expressed using \(2\theta\).

Example 4: Reflection About a Line

Find the reflection of the vector \(\mathbf{x} = (1,5)\) about the line through the origin that makes an angle of \(\pi /6\) (\(30^{\circ}\)) with the \(x\)-axis.

Here, we use the reflection matrix formula directly. Notice that the angle used in the matrix is \(2\theta\), not \(\theta\). This is a common point of confusion, so be careful. The calculation shows the coordinates of the point \((1,5)\) reflected across the line at \(30^\circ\). This is another fundamental transformation with applications in computer vision and graphics.

Key Takeaways

• Reflections: Mirror images across lines/planes.
• Projections: “Shadows” onto lines/planes.
• Rotations: Circular movement around an origin/axis.
  • Crucial for ECE applications like Yaw, Pitch, Roll.
• Dilations/Contractions: Uniform scaling of objects.
• Expansions/Compressions: Non-uniform scaling along specific axes.
• Shears: Tilting or slanting objects.

Each transformation has a unique standard matrix. Understanding these matrices is fundamental to linear algebra and its applications.

We’ve covered a lot of ground today, from simple reflections to more complex shears. The key message is that each of these geometric transformations can be represented by a matrix, and applying the transformation simply involves multiplying a vector by that matrix. This matrix representation is what makes linear algebra so powerful for engineers. Review the interactive examples and try to connect the visual changes with the mathematical operations.