Suppose that x1=-1, x2=2, x3=4, x4=-3 is a solution of a...

Suppose that $x_{1}=-1, x_{2}=2, x_{3}=4, x_{4}=-3$ is a solution of a nonhomogencous linear system $A x=b$ and that the solution set of the homogeneous system $A \mathbf{x}=\mathbf{0}$ is given by the formulas. $$ x_{1}=-3 r+4 s, \quad x_{2}=r-s, \quad x_{3}=r, \quad x_{4}=s $$ (a) Find a vector form of the general solution of $A \mathbf{x}=\mathbf{0}$. (b) Find a vector form of the general solution of $A \mathbf{x}=\mathbf{b}$

Suppose that $x_{1}=-1, x_{2}=2, x_{3}=4, x_{4}=-3$ is a solution of a nonhomogencous linear system $A x=b$ and that the solution set of the homogeneous system $A \mathbf{x}=\mathbf{0}$ is given by the formulas.
$$
x_{1}=-3 r+4 s, \quad x_{2}=r-s, \quad x_{3}=r, \quad x_{4}=s
$$
(a) Find a vector form of the general solution of $A \mathbf{x}=\mathbf{0}$.
(b) Find a vector form of the general solution of $A \mathbf{x}=\mathbf{b}$

Key Concepts

Nonhomogeneous Linear Systems

A nonhomogeneous linear system is one where the system of equations is of the form Ax = b, with b being a non-zero vector. The presence of a non-zero b distinguishes it from homogeneous systems, and such systems typically have either a unique solution or infinitely many solutions—depending on the consistency of the system and the properties of the matrix A.

Homogeneous Linear Systems

A homogeneous linear system is one where the system of equations is of the form Ax = 0. Every homogeneous system is always consistent since the zero vector is a solution, and its solution set forms a subspace, often described in terms of free parameters indicating directions in which the solutions can vary.

Particular Solution

A particular solution to a nonhomogeneous linear system is a specific solution that satisfies Ax = b. It is important because it provides a specific 'base' solution, and once it is known, the general solution to the system can be constructed by adding the general solution of the corresponding homogeneous system.

General Solution and the Superposition Principle

The general solution of a linear system combines the particular solution of the nonhomogeneous system and the general solution of the homogeneous system (or null space). This principle is based on the fact that if x_p is a particular solution of Ax = b and x_h is any solution of Ax = 0, then x_p + x_h is a solution of Ax = b. This superposition is a fundamental property of linear systems.

Vector Form of the Solution Set

Expressing the solution set in vector form involves writing the solution as a sum of a fixed vector (the particular solution) plus a linear combination of vectors that span the null space of A. This form is particularly useful because it clearly shows the structure of the solution space and how parameters come into play, making it easier to understand and analyze the set of all possible solutions.

Transcript

Please give Ace some feedback