A system of linear equations is written in matrix-vector form as Ax = b.
Understanding the Matrix-Vector Form: Ax = b
As stated in the reference, a system of m linear equations in n unknowns can be compactly represented using matrices and vectors. This representation is particularly useful for analyzing and solving systems of equations, especially large ones, and is fundamental in fields like linear algebra, engineering, and computer science.
The standard matrix-vector form for such a system is:
Ax = b
Where:
- A is the coefficient matrix of the system.
- x is the vector consisting of the unknowns.
- b is the vector consisting of the right-hand sides of the equations.
Components of Ax = b
Let's break down each part of the matrix-vector form:
The Coefficient Matrix (A)
This matrix contains all the numerical coefficients of the variables from the system of equations.
- If the system has m equations and n unknowns, the coefficient matrix A will have dimensions m × n.
- Each row of A corresponds to one equation, and each column corresponds to one unknown variable.
- The entry in the i-th row and j-th column of A is the coefficient of the j-th variable in the i-th equation.
The Vector of Unknowns (x)
This is a column vector that lists all the variables (unknowns) in the system.
- Since there are n unknowns, the vector x will be an n × 1 column vector.
The Vector of Right-Hand Sides (b)
This is a column vector containing the constant terms from the right side of each equation in the system.
- Since there are m equations, the vector b will be an m × 1 column vector.
Example: Converting a System to Matrix-Vector Form
Let's consider a simple system of two linear equations with two unknowns:
Equation 1: 2x + 3y = 7
Equation 2: x - y = 1
Here, m = 2 (two equations) and n = 2 (two unknowns, x and y).
To write this system in the form Ax = b, we identify A, x, and b:
- A (the coefficient matrix): The coefficients for the first equation are 2 and 3. The coefficients for the second equation are 1 and -1.
- x (the vector of unknowns): The unknowns are x and y.
- b (the vector of right-hand sides): The right-hand sides are 7 and 1.
Putting it together:
| Component | Matrix/Vector Representation | Notes |
|---|---|---|
| Coefficient Matrix A | $\begin{bmatrix} 2 & 3 \ 1 & -1 \end{bmatrix}$ | (2 × 2 matrix) |
| Vector of Unknowns x | $\begin{bmatrix} x \ y \end{bmatrix}$ | (2 × 1 column vector) |
| Vector of Right-Sides b | $\begin{bmatrix} 7 \ 1 \end{bmatrix}$ | (2 × 1 column vector) |
So, the matrix-vector form of the system $\begin{cases} 2x + 3y = 7 \ x - y = 1 \end{cases}$ is:
$\begin{bmatrix} 2 & 3 \ 1 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 7 \ 1 \end{bmatrix}$
This representation precisely matches the Ax = b structure described in the reference, where A is the 2x2 coefficient matrix, x is the 2x1 vector of unknowns, and b is the 2x1 vector of right-hand sides.
