Problem 54 - College Algebra

Question

The figure shows the letter \(L\) in a rectangular coordinate system. (GRAPH CANNOT COPY) The figure can be represented by the matrix $$B=\left[\begin{array}{llllll}0 & 3 & 3 & 1 & 1 & 0 \\\0 & 0 & 1 & 1 & 5 & 5\end{array}\right]$$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The \(L\) is completed by connecting the last point in the matrix, \((0,5),\) to the starting point, \((0,0) .\) Use these ideas to solve Exercises \(53-60 .\) Use matrix operations to move the L 2 units to the right and 3 units down. Then graph the letter and its transformation in a rectangular coordinate system.

Step-by-Step Solution

Answer

The transformed matrix obtained after moving the 'L' 2 units to the right and 3 units down is given by [[2, 5, 5, 3, 3, 2], [-3, -3, -2, -2, 2, 2]].

Step 1: Understanding the Given Matrix and Transformation

The given matrix B is a 2x6 matrix that represents an 'L' shape. Each column in this matrix corresponds to a point that forms the 'L' shape. Our task is to move this 'L' shape 2 units to the right and 3 units down. In terms of a matrix, this corresponds to adding 2 to each x-value (which is each number in the first row of the matrix) and subtracting 3 from each y-value (which is each number in the second row of the matrix).

Step 2: Performing the Transformation

To perform this transformation, we will create a matrix T that represents the transformation and add it to our original matrix B. The matrix T will have the same dimensions as B and will be filled with the transform values - in this case, 2 for x (first row) and -3 for y (second row). Thus, we get \(T = [[2, 2, 2, 2, 2, 2], [-3, -3, -3, -3, -3, -3]]\). Adding B and T together will give us a new matrix representing the transformed 'L'.

Step 3: Getting the Transformed Matrix

Adding matrices is straightforward – we simply add corresponding elements together. So, we add matrix T to matrix B (that is, each element of T is added to the corresponding element of B): \(B+T = [[0+2, 3+2, 3+2, 1+2, 1+2, 0+2], [0-3, 0-3, 1-3, 1-3, 5-3, 5-3]] = [[2, 5, 5, 3, 3, 2], [-3, -3, -2, -2, 2, 2]]\). This new 2x6 matrix represents the transformed 'L' shape.

Key Concepts

Coordinate System

A coordinate system is a method for identifying the precise location of points in a space. The most common coordinate system is the rectangular or Cartesian coordinate system, which uses two perpendicular lines known as axes. The point where these axes intersect is called the origin, typically denoted as (0, 0) in a 2D space.

• The x-axis runs horizontally and is used to measure left and right positions.
• The y-axis runs vertically and measures up and down positions.

Coordinates are written in the form of an ordered pair (x, y), where x represents the horizontal displacement from the origin, and y represents the vertical displacement. This system helps to graphically represent shapes or figures, like the letter 'L' from the exercise, on a flat plane.

Matrix Addition

Matrix addition is one of the fundamental operations you can perform on matrices. When adding matrices, it's essential that the matrices are of the same dimension, meaning they have the same number of rows and columns. This allows for element-wise addition, where the corresponding elements from each matrix are added together.
Here's how matrix addition works:

• Consider two matrices, A and B, both of which are m x n in dimension.
• To add them, simply add each element of A with the corresponding element of B.
• The result is a new matrix of the same dimensions.

In our exercise, this process helps to shift the points of the 'L' shape, by adding a transformation matrix T to the original matrix B. This shifts the entire shape in the intended direction.

Transformation Matrix

A transformation matrix is a tool used to apply specific changes to geometric figures, such as translations, rotations, or scalings. In the context of our exercise, the transformation matrix T is used to translate, or move, the shape of the letter 'L' to a new position in the coordinate plane.

• Translation involves shifting every point of a shape by the same amount in specific directions.
• The matrix T in the example is defined as T = \( \begin{bmatrix} 2 & 2 & 2 & 2 & 2 & 2 \ -3 & -3 & -3 & -3 & -3 & -3 \end{bmatrix} \).
• This matrix effectively moves each point in the original matrix B by adding 2 to every x-coordinate (shifting right) and subtracting 3 from every y-coordinate (shifting down).

Transformation matrices allow for efficient manipulation of figures, making it easy to calculate new positions without altering the shape's size or orientation.

Geometric Transformations

Geometric transformations include a variety of methods used to modify the position, size, and orientation of shapes. These transformations are fundamental in fields such as computer graphics and robotics to simulate movements or adjustments in shapes.
There are several types of geometric transformations:

• **Translations**: Moves the entire shape a certain distance in a specified direction. It's a straightforward modification that does not alter the shape's form. In our example, the 'L' is translated horizontally by +2 and vertically by -3.
• **Rotations**: Spins a shape around a specific point, called the pivot point, by a certain angle.
• **Scalings**: Resizes the shape, either enlarging or shrinking, while maintaining its proportions.
• **Reflections**: Flips the shape over a line, creating a mirror image.

Understanding geometric transformations helps in recognizing how figures can interact with their environments and how they can be manipulated using matrices.