Question

A list of transformations is given. Find the matrix \(A\) that performs those transformations, in order, on the Cartesian plane. (a) horizontal shear by a factor of 2 (b) vertical shear by a factor of 2

Short answer

The matrix is \( \begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix} \).

Step-by-step solution

Understand Horizontal Shear Transformation

A horizontal shear transformation moves each point horizontally in proportion to its vertical distance from the x-axis. A shear by a factor of 2 means every vertical unit is shifted 2 units horizontally. The matrix for a horizontal shear with factor 2 is: \[ \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} \].

Understand Vertical Shear Transformation

A vertical shear transformation moves each point vertically in proportion to its horizontal distance from the y-axis. A shear by a factor of 2 means every horizontal unit is shifted 2 units vertically. The matrix for a vertical shear with factor 2 is: \[ \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix} \].

Combine Transformations

When performing multiple transformations, the transformation matrices are multiplied in reverse order of application. First apply the horizontal shear, then the vertical shear. Therefore, multiply the horizontal shear matrix by the vertical shear matrix.Compute the product:\[ \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 \times 1 + 0 \times 2 & 1 \times 2 + 0 \times 1 \ 2 \times 1 + 1 \times 0 & 2 \times 2 + 1 \times 1 \end{bmatrix} \]\[ = \begin{bmatrix} 1 & 2 \ 2 & 5 \end{bmatrix} \].

Resultant Matrix

The matrix that performs a horizontal shear by factor 2 followed by a vertical shear by factor 2 is \[ \begin{bmatrix} 1 & 2 \ 2 & 5 \end{bmatrix} \].

Key concepts

Horizontal Shear

A horizontal shear transformation changes the x-coordinate of a point on the Cartesian plane while maintaining its y-coordinate unchanged. This transformation effectively means that the image will stretch or compress horizontally depending on the shear factor. If the shear factor is 2, every unit movement in the vertical direction causes each point to move 2 units horizontally. The transformation matrix representing this operation is given by:

• Matrix form: \[ \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} \]
• 1 remains as the scaling factor along the x and y axes, and 2 is applied as a translation along the x-axis for each unit moved vertically.

This alignment effectively skews objects by making vertical lines tilt to the right, visualizing a stretching effect along the x-axis. It is crucial to multiply the shearing factor correctly to ensure intended transformations.

Vertical Shear

Vertical shear involves changing the y-coordinate of any given point on the Cartesian plane while keeping its x-coordinate steady. Imagine pushing the top and bottom of a shape in opposite vertical directions - that's the effect of vertical shear. Here, if the shear factor is 2, it means that for every unit distance along the x-axis, the point is moved 2 units up or down.

• The transformation matrix form used is:\[ \begin{bmatrix} 1 & 0 \ 2 & 1 \end{bmatrix} \]
• 1 continues as the base scaling coefficient for both axes while 2 facilitates vertical translation depending on x-coordinates.

In practical terms, the transformation tilts horizontal lines upward, creating a stretching effect vertically. When applied correctly, vertical shear can visually elongate shapes or skew them to simulate perspective or distortion effects.

Matrix Multiplication

Matrix multiplication is a process used to combine several transformations into one. This is especially useful in 2D graphics to apply complex transformations efficiently. To apply transformations like horizontal and vertical shear to an object on the Cartesian plane, their matrices must be multiplied together in a specific order.

• Order matters: the matrix representing the last transformation is on the left of the multiplication product.
• Step by step, for \[ A \times B = C \]: The element at position \( (i,j) \) in the resulting matrix \( C \) is the sum of the products of the elements of row \( i \) in \( A \) and column \( j \) in \( B \).

This method ensures the sequence of transformations is maintained accurately, consolidating the separate transformations into a single matrix that can be applied to any point on the Cartesian plane.

Cartesian Plane

The Cartesian plane is a standard grid formed by an x-axis and y-axis that double as reference lines for locating points. Understanding its layout is essential for applying matrix transformations effectively. Points on the plane are defined by coordinates \( (x, y) \), describing their horizontal and vertical distances from the origin, where the axes intersect.

• Essentially a 2D grid: x-axis indicates horizontal position, y-axis indicates vertical position.
• Facilitates graphical transformations: changes are evident when transformation matrices are applied to these coordinates.

When transforming shapes or figures using matrices, the Cartesian plane provides the framework to visualize and calculate their new positions. It's a fundamental element in geometry and graphics design, allowing for a clear representation of linear transformations.

Transformation Matrices

Transformation matrices are crucial tools in linear algebra. They help achieve specific changes or transformations on figures in the Cartesian plane. Whether you want to rotate, scale, translate, or shear objects, transformation matrices make it possible by altering the coordinates through matrix multiplication.

• Each transformation type has its unique matrix form, such as identity, rotation, translation, etc.
• Combining transformations: achieved by multiplying respective matrices together in the proper sequence, handling multiple transformations simultaneously.

The real power of transformation matrices lies in their ability to simplify complex geometric manipulations mathematically. Whether in computer graphics, physics simulations, or solving engineering problems, these matrices make handling transformations efficient and versatile. By using them, transformations can be applied accurately and understood in terms of mathematical relationships between points and shapes.