# Linear Algebra - (Linear system|System of Linear equations)

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.

\begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}

Each linear system corresponds to a linear system with zero right-hand sides:

• $a_1 . x = beta_1 \rightarrow a_1 . x = 0$
• $a_2 . x = beta_2 \rightarrow a_2 . x = 0$
• $a_n . x = beta_n \rightarrow a_n . x = 0$

If a linear system has a solution u1 then that solution is unique if the only solution to the corresponding homogeneous linear system is 0.

## Type

### Homogeneous

A system of Linear Algebra - Linear Equation is called a homogeneous linear system.

\begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 0 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& 0 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}

The solution set of a homogeneous linear system is a vector space.

Lemma: Let u1 be a solution to a linear system. Then, for any other vector u2, u2 is also a solution if and only if u2 - u1 is a solution to the corresponding homogeneous linear system

### Triangular

A triangular linear system as a triangular form

• $10 = 2x_1 + 3x_2 - 4x_3$
• $~3 = 1x_2 + 2x_3$
• $15 = 5x_3$

and can be expressed as a triangular matrix

## Documentation / Reference

Discover More
Linear Algebra - (Gaussian|Common) Elimination

Method illustrated in Chapter Eight of a Chinese text, The Nine Chapters on the Mathematical Art, that was written roughly two thousand years ago. Rediscovered in Europe by Isaac Newton (England) and Michel...
Linear Algebra - Linear Equation

A linear equation represents a linear function that forms a straight line. A common form of a linear equation in the two variables (two dimensions) x and y is where: m is a constant named the slope...
Linear Algebra - Matrix Equation

A linear vector equation is equivalent to a matrix equation of the form : where: A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries. A matrix...
Linear Algebra - Null Space of a (Matrix|Vector Space)

Null space of a matrix A (Written Null A) is: The Null space of a matrix is a basis for the solution set of a homogeneous linear system that can then be described as a homogeneous matrix equation. A...
Linear Algebra - Triangular Matrix

The matrix is a triangular matrix. Definition: An n x n upper triangular matrix A is a matrix with the property that . The entries forming the triangle can be be zero or nonzero. We can use backward substitution...