Linear Algebra Course Projects

Here is the explanation about three projects of Linear Algebra course:

• LU decomposition
• Least Squares based Signal Denoising
• Using SVD decomposition for image compression

LU decomposition

The solution of Ax=b is done with Row Reduction in ${O(n^3)}$. But if we have the LU decomposition of the matrix, the solution of the equation can be done in ${O(n^2)}$. We also know that the calculation of the matrices L and U itself takes place in time ${O(n^3)}$. So that, solving a linear equation system by calculating the LU decomposition of its matrix and finding the solution of the device from it is not a effective method. Now we want to solve a large number of equations in the form of Ax=b in which A is constant and only b is different, we can solve the equation for different b's in ${O(n^2)}$ time, which is much more optimal.

Least Squares based Signal Denoising

The diagram shows the price of Bitcoin every 2 hours from the end of 2020 to the 20th of May. Suppose that the vector 𝒚 is the vector of bitcoin price values, the unknown vector 𝒙 is the noise-free vector of the price we are looking for, and the vector 𝒗 is the uncertain noise vector. That is, we have: ${y=lx+v}$

The results of denoising would be like this:

   
         λ=10; not denoised                λ=100; well denoised                 λ=10000; too denoised 

Using SVD decomposition for image compression

We can decompose a given image into the three color channels red, green and blue. Each channel can be represented as a (m × n)‑matrix with values ranging from 0 to 255. We will now compress the matrix A representing one of the channels. To do this, we compute an approximation to the matrix A that takes only a fraction of the space to store. Now here's the great thing about SVD: the data in the matrices U, Σ and V is sorted by how much it contributes to the matrix A in the product. That enables us to get quite a good approximation by simply considering only the k-terms of the first important parts of the matrices

   
              k=50;                               k=250;                               k=750;