Question

By multiplying matrices, find \(x^{\prime \prime}, y^{\prime \prime}\) in terms of \(x, y\), given: $$ \left\\{\begin{array} { l } { x ^ { \prime } = 2 x + 5 y } \\ { y ^ { \prime } = x + 3 y } \end{array} \quad \left\\{\begin{array}{l} x^{\prime \prime}=x^{\prime}-2 y^{\prime} \\ y^{\prime \prime}=3 x^{\prime}-5 y^{\prime} \end{array}\right.\right. $$ and say what change of axes has been made.

Short answer

x'' = -y and y'' = x. The change of axes is a 90-degree counterclockwise rotation.

Step-by-step solution

Represent equations as matrices

The given system of equations can be written in matrix form. First, write the equations for \( x' \) and \( y' \): \[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 2 & 5 \ 1 & 3 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]

Write equations for x'' and y''

The equations for \( x'' \) and \( y'' \) are given. We will use the values of \( x' \) and \( y' \) from the previous step to substitute in these equations. \[ x'' = x' - 2y' \] \[ y'' = 3x' - 5y' \]

Substitute x' and y' into x''

Replace \( x' \) and \( y' \) in the \( x'' \) equation. \[ x'' = (2x + 5y) - 2(x + 3y) \] \[ x'' = 2x + 5y - 2x - 6y \] \[ x'' = -y \]

Substitute x' and y' into y''

Do the same for \( y'' \). \[ y'' = 3(2x + 5y) - 5(x + 3y) \] \[ y'' = 6x + 15y - 5x - 15y \] \[ y'' = x \]

Find matrix form for relation

Now, represent \( x'' \) and \( y'' \) in matrix form. \[ \begin{pmatrix} x'' \ y'' \end{pmatrix} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]

Determine the change of axes

The transformation matrix \[ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \] is a rotation matrix representing a 90-degree counterclockwise rotation of the coordinate axes around the origin.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices to generate a new matrix. The multiplication is done by taking the dot product of rows and columns from the two matrices.
In the exercise, we express the transformations as matrix equations:
\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 2 & 5 \ 1 & 3 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
For each new element, we multiply corresponding elements and sum them up:
\[ x' = 2x + 5y \] \[ y' = x + 3y \]
Remember to align the rows of the first matrix with the columns of the second matrix.

Coordinate Transformation

Coordinate transformation is the changing of coordinates from one system to another. In mathematics, it's common to use matrices for this.
In the given exercise, we start with old coordinates \[ (x, y) \] and transform them first to \[ (x', y') \], then to \[ (x'', y'') \].
Transforming \[ (x', y') \]:
\[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 2 & 5 \ 1 & 3 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
Transforming \[ (x'', y'') \]:
\[ \begin{pmatrix} x'' \ y'' \end{pmatrix} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
This means that coordinates have been transformed by specific matrix operations.

Rotation Matrix

A rotation matrix is used to rotate points in the coordinate plane. It is a square matrix that represents a rotation.
The general form of a 2D rotation matrix by angle \[ \theta \] is:
\[ \begin{pmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \]
In our exercise, the rotation matrix used is:
\[ \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \]
This particular matrix represents a 90-degree counterclockwise rotation. The mathematical operations reflect this specific transformation.
Therefore, after applying this matrix, we move our axes by 90 degrees counterclockwise.

System of Linear Equations

A system of linear equations is a set of equations with multiple variables that you solve together. Each equation represents a line, and the solution is where lines intersect.
In the exercise, we start with two equations:
\[ \begin{array}{l} x' = 2x + 5y \ y' = x + 3y \ \end{array} \]
We transform further:
\[ \begin{array}{l} x'' = x' - 2y' \ y'' = 3x' - 5y' \ \end{array} \]
Solve the system step by step to find relationships for \[ x'' \] and \[ y'' \] in terms of \[ x \] and \[ y \]. Each step involves substitution and simplification, accurately transforming between coordinate systems.