A norm of linear transformation \(\Lambda : X \rightarrow Y\) is defined by \(\| \Lambda \| = \sup \{ \| \Lambda(x) \|:x \in X, \|x\| \le 1 \}\). We can give a norm to a space or a set. A norm determines the size of a vector in the function space. The way of measure the size of a vector gives important properties of the space like boundness, completeness or orthogonality.

It is the main subject of analysis that finding conditions making sequential mathematical objects like a set, sequence, series to be convergent. Induction changes \(S = \mathbb{N}\) to \(s_{1} \in S\) and if $ s_{n} S$ then \(s_{n+1} \in S\). The natural number has a property of endless addable with one. But, induction can prove only natural number \(\mathbb {N}\) not infinity \(\infty\).
\[ Induction \\ s_{1} \in S \\ if\ s_{n} \in S \ then \ s_{n+1} \in S \\ Then\ S = \mathbb{N} \\ \] The limit is the way \(\mathbb {N}\) goes to \(\infty\).

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 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.

The optimization problem have two components that are objective function \(f_0 : \mathbb R ^n \rightarrow \mathbb R\) and the constraints. The objective function and constraints keep in check each other and make balance at saddle point i.e. optimal point. The dual (Lagrange) problem of the optimal problem also solve the optimization problem by making low boundary.
The dual problem can be explained as a conjugate function \(f^* = \sup (x^Ty-f(x))\).

The purpose of approximation is finding optimal point \(x^*\) i.e. \(\nabla F(x^*) = 0\). We need a step/search direction \(\Delta x\) and step size \(t\). Taylor approximation has polynomial arguments that is a step and parameters of derivatives at the start point. The first degree of Taylor approximation has one adding term from start point \((x_0, F(x_0))\). The adding term \(\nabla F(x) \Delta x\) is consistent with a parameter (gradient \(\nabla F(x)\)) and a argument (step \(\Delta x\)).

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.

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.

The meaning of \(A^{T}\)
Steady state equilibrium Graph Laplacian matrix \(A^{T}CA\) Differential equation and Laplacian matrix Derivative is a graph without branch. Row space and column space are dual. \(A\) and \(A^{T}\) are dual. ref) Linear algebra and learning from data, Part IV, Gilbert Strang

Differential equations describe the change of state. The change relates to the state. The solutions of the differential equations are the status equations. The initial conditions set the time \(t\) and status \(y\). The boundary conditions are the value of boundary \(y_0\) and \(y_1\).
\(dy \over dt\) \(= ay + q(t)\) starting from \(y(0)\) at $t=0. inital conditions \(t = 0\) and \(y=1\)
\(q(t)\) is a input and \(y(t)\) is a response.

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