Systems of linear equations

Let us have a linear system of

$$\sum _{j=1}{a}_{ij}{x}_j=b_i$$

where we have $m$ equations about $n$ variables. One can construct a matrix form as the following

$$A_{m\times n}{x}_{n\times 1}=b_{m\times 1}$$

Definition

We call

A|b \end{bmatrix} $$ the **augmented form** for the linear system.

Solutions of square systems

Suppose that $A\in R^{n\times n}$. First we reduce the $A$ to row echelon form using Gaussian elimination.

There are two scenarios

1. Some columns do note have leading $1$‘s $⟹$ there are free variables
2. All columns have leading $1$‘s $⟹$ system can be reduced into $I_n$ $⟹$ there are no free variables.

Theorem

A matrix $A$ is invertible $⟺$ it can be row reduced into identity matrix.

Is the system is invertible then there exists a unique solution

$$A^{-1}Ax=A^{-1}b⟹x=A^{-1}b$$

Now, if the system is not invertible we have two possibilities

1. There are no solutions since a contradictory equation of $0=c$, $c\ne 0$.
2. There are infinitely many solutions when there aren’t any contradictory equations.

If the system is homogeneous, meaning that $b=0$ then there is only two possibilities:

1. $A$ is invertible $⟺Ax=0$ has a unique solutions, namely $x=0$, which is the trivial solution
2. $A$ is not invertible $⟺Ax=0$ has infinitely many solutions ($\det A=0$). Then the system is said to have non-trivial solutions.