Enters matrix

Enters matrix

Problem: In linear algebra, a linear transform is a mapping of an m-dimensional vector into an n-dimensional vector. Familiar examples of linear transformations include rotation and scaling of a vector in the plane (a mapping from a 2-dimension vector to a 2-dimensional vector), projection of a 3-dimensional vector into the plane (a mapping from a 3-dimension vector to a 2-dimensional vector), and so on.

Linear transformation for n-dimensional vectors into m-dimensional vectors is represented by a matrix of m-rows and n columns, or an m x n matrix. Here are some examples from Wikipedia; look under the heading “Linear transformations”.

Let M be an mxn matrix and V be an n-dimensional vector. The m-dimensional vector U = {u_1, u_2,…, u_m} obtained by applying M to V can be calculated by multiplying the matrix M and the vector V:

U = MV,

Where U_i = M_i  ·  V, M_i is the vector of the i-th row of matrix M and  · is the inner product operator, i = 1, …, m.

For example,

A program is needed to compute the vector U, the linear transformation of a given vector V and a matrix M. Specifically, prompt the user to input the name of a file that contain a matrix with the following format:

m n

elements of row 1

…

elements of row m

For the matrix above, the input file looks like this:

2 3

1 2 3

4 5 6

Then, prompt the user to input a vector V of the following format from the console

n [V_1, V_2, …, V_n].

When this is done, the program shall compute the vector U when it is defined, or displays an error message when it is not defined. Prompt the user whether to continue or stop.

In addition to the functional requirements above, the program must meet the constraint of using class, functions, and their tests from the project Inner Product project.

Step U. Understanding the problem.

Functionally, the program is similar to the inner product computation. Of course, we will need a new class to represent matrix, and functions for computing linear transformation and doing file I/O. By now, we have some experiences. We do need to consider the constraints of asking us to use the units from inner product computation.

Step D. Devising a plan.

Here are the tasks.

T1. Prepare two projects, Linear Transformation and it test project.

T2. Write up the Matrix class with the member functions to get row vectors and elements.

T3. Write unit tests for Matrix.

T4. Write file I/O for Matrix.

T5. Write tests for file I/O for Matrix.

T6. Write main function.

Step C. Carrying out the plan.

Let’s start with T1. While the normal way to reuse artifacts from the inner product project is to create a library and use it from the current projects, for simplicity, we will just copy the source files from the inner project into the linear transformation projects and it test projects.

The tasks T2 to T5 should pose little challenge now, we will leave it to you as an exercise to complete them. As before, I strongly encourage you to write up a unit test before you do the member function or function. This has the effect of keeping the class and functions small. However, beware of the memory management issue. Since memory from heap will be used, rather than accepting the C++ defaults, you should write the copy constructor, the destructor, and the assignment operator.

Step L. Looking back.

We see that due to the separation of production and test projects, we can now move at a steady pace by growing the code for matrix piecemeal, all the while insisting that all unit tests pass.

Arguably, the present problem is no less simple than the problem of computing inner products for two vectors. We are able to handle it as a single problem, thanks to our experience from solving the inner product problem. Our capability to handle problem grows. 


© Y C Cheng, 2013, 2014. All rights reserved.