Here is the explanation about three projects of Linear Algebra course:
- LU decomposition
- Least Squares based Signal Denoising
- Using SVD decomposition for image compression
The solution of Ax=b is done with Row Reduction
in
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:
The results of denoising would be like this:
λ=10; not denoised λ=100; well denoised λ=10000; too denoised
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;