linear system

linear system is a system of linear equations, which can be represented by an augmented matrix

where is the square matrix of coefficients, is the column vector of unknowns and is the column vector of constant

A solution of a linear system is a set of variables satisfy ALL of linear equations.

If it has no solutions, we say the system is inconsistent.

Example

Tickets fora movie at the local cinema costs 15$20$1800$, how many of each ticket was sold?

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linear system:

augmented matrix:

# sage math
# 2x3 matrix of real numbers
matrix(RR, [[1, 2, 3],[4, 5, 6]])
 
# 2x3 matrix of rational numbers
matrix(QQ, [[1, 2, 3],[4, 5, 6]])

Relationship with fundamental subspaces

Consider the linear system

The system is consistent system if and only if we can find the solution such that

The left-hand side forms the column space. So the solution only exists if is in the column space .

Approach 1 with column space

Strategy to determine if the system (where is consistent

1. Find the basis for column space
2. Determine the dimension of this column space & dimension of
3. If , the column space spans and thus contains . The system is consistent

Approach 2 with left null space

Strategy to determine if the system (where has no solution

Check if is orthogonal to the left nullspace

1. transpose
2. Reduce to rref and find the parametric solution
3. Find the basis for left nullspace
4. Compute the dot product of every vector in the column space of with every vector in the basis above. If all results are zero, the linear system has no solutions.

Examples