# Gilbert Strang

## Laplace transformation

The Fourier series represents a periodic function as a descrete vectors. The Fourier transformation turns a time domain non-periodic function into a frequency domain continuous function. The Fourier series and transformation change a single time base $$t$$ into infinite frequency basis $$e^{inx}$$ or $$e^{iwx}$$. The function on infinite basis domain can be represented by a vector or a function of basis domain $$v_{n}$$ or $$f(w)$$. This is a coefficients of Fourier series or Fourier transformation.

## Convolution and Fourier transformation

Convolution is a vector operation on two vectors. $Convolution \\ c * d = d*c \\ (c*d)_n = \Sigma_{i+j} c_i d_j = \Sigma_i c_i d_{n-i}.$ This is multiplying polynomials. The parameters of multiplied polynomial become convolution of two polynomials. Fourier transformation expands x base to infinite exponential basis $$e^{iwk}$$. The multiplication on x (time) space becomes convolutionn on k (frequency) space. If time space is periodic, its Fourier transformation is discrete i.

## Singular vector decomposition

Bases are the central idea of linear algebra. An invertable square matrix has eigenvectors. A symetric matrix has orthogonal eigenvectors with non-negative eigenvalues, i.e. positive semidefinite. A matrix has two types of singular vectors, left and right signular vectors, $$A=U\Sigma V^{T}$$. When we think the matrix $$A$$ is data points of rows $$A=U\Sigma V^{T}$$ like data table, The right singular vectors $$V$$ build bases, the sigular values $$\Sigma$$ are magnitude of the bases and the left singular values $$U$$ becomes new data points on new bases.

## Low rank matrix and compressed sensing

This is a note for part III of Linear Algebra and learning from data, Gilbert Strang The main themes are sparsity (Low rank), Information theory (compression), and of course linear transformation. A full rank matrix is inefficient. Finding low lank matrix which is close with original matrix can save computation. The rank one matrix $$uv^{T}$$ is a unit of a matrix. The full rank matrix can be decomposed by sum of rank one matrices i.