Exercise 1 (Strings, Arrays, Classes, Methods)

• Let \( f\colon \mathbb{R}^n\rightarrow \mathbb{R}^m \) be a linear map.
• Compute the transformation matrix \( M \) for arbitrary \( f \) such that \( f(\vec{x})=M\vec{x} \) for all \( \vec{x}\in\mathbb{R}^n \).
  

Assumptions

• We approximate \( x\in \mathbb{R} \) by float values and vectors \( \vec{x} \) by float arrays.
• A linear function \( f \) is represented by a class named LinearFnc.
• Assume that LinearFnc contains a method applyTo which maps a float array of length \( n \) to a float array of length \( m \).

public class LinearFnc {<br>    ...<br>    public float[] applyTo(float[] domVec) {<br>        float[] ranVec = new float[m];<br>        ...<br>        return ranVec;<br>    }
}

Implementation

• Write an arbitrary test class which contains the entry point public static void main(...) {...}. This class is for testing purpose only and must not be submitted.
  
• Write a class Matrix which represents \( M \) in a separate file Matrix.java. This class should not contain an entry point public static void main(...) {...}.
• The constructor of Matrix fixes the dimension \( m\times n \).
• The class Matrix should contain three methods:

public void initMatrix(LinearFnc lfc) {...<br>public float[] applyTo(float[] domVec) {...<br>public String toString() {...<br></pre></div></div><ul><ul><li><b>initMatrix</b> computes the internal representation of \( M \), such that \( M\vec{x} \) can be computed without \( f \), for arbitrary \( \vec{x}\in\mathbb{R}^n \). The internal representation should be private.
applyTo computes \( M\vec{x} \), such that \( f(\vec{x})=M\vec{x} \) for all \( \vec{x}\in\mathbb{R}^n \).
toString returns a String representation of the transformation matrix \( M \).
Submit only the file Matrix.java.
Hint
Implement a class LinearFnc which represents some fixed \( f \) and compare the result of Matrix.applyTo with the result of LinearFnc.applyTo for some test cases.