Refresh on Linear Algebra

#Preface

Data can be described as the collection of variables. We collect the data as label it as X and Y. For example, we can measure the rainwater and temperature. X can be the temperature. Y can be the amount of rainwater. We can then attempt to map a relationship between X and Y. $y = mx + b$ We need to figure out what the m and b are. These coefficients and bias are unknown and when we are given a large amount of data can we suggest a reasonable value for both.

Now imagine, we collect multiple variables not just temperature for X. X1 can be the region. X2 can be the amount of humans in the area. Now the equation looks like this $y = m_1x_1 + m_2x_2 + m_3x_3 + b_1 + b_2 + b_3$ This can be rewritten in matrix form as: $y = m*x+b$ m is a 1x3 row vector and x is a 3x1 column vector

Now if there are multiple equations you have a system of equations. In this case, we have multiple variables. We have multiple constraints. We model this as a system of equations. In solving this system of equations can we arrive at a possible solution should such a solution exist. $Ax=0$

This matrix A is known as the coefficient matrix.

#System of equations A system can be described as consistent or inconsistent.

consistent has at least one solution
inconsistent has no solution Checked if the augmented matrix has a contradictory row

A system can be described as having a unique or nonunique solution or no solution

A system can be said to be linear or nonlinear.

The rank of the system is the number of linearly independent rows or columns. If rank equals the number of unknowns, there is a unique solution

The determinant of the sytem if it is nonzero implies a unique solution

#Matrix fundamentals

Given a matrix A, we can solve for the null space Ax=0.

A matrix can have rows dependent on other rows. Row Dependence can be detected via row reduction (Guassian Elimination). A matrix has four fundamental subspaces.

A matrix A, $A \in \mathbb{R}^{m*n}$
Column space C(A), dim = r $Col(A) = \{Ax | x\in \mathbb{R}^n \}$
Row space, C(A Tranpose), dim=r $Row(A) = Span\{ rows of A\}$
(Right) Null space N(A), dim=n-r,
Left Null space N(A Transpose), dim=m-r

#Symbolic

\text{Given A is the coefficient matrix, it has dimensions m rows and n columns thus this matrix exists in the R^{mxn} space.} \newline \text{Now consider the input vector x this vector MUST be of dimensions m and exist in R^m space.} \newline \text{We know that a system of equations is setup to define what MUST be true in order for the system we are modeling to exist. In other words the row space is a subspace of the total space of all x vectors but you should not }