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Chapter 1: Problem 47

A linear system of the form $$A \vec{x}=\overrightarrow{0}$$ is called homogeneous. Justify the following facts: a. All homogeneous systems are consistent. b. A homogeneous system with fewer equations than unknowns has infinitely many solutions. c. If \(\vec{x}_{1}\) and \(\vec{x}_{2}\) are solutions of the homogeneous sys\(\operatorname{tem} A \vec{x}=0,\) then \(\vec{x}_{1}+\vec{x}_{2}\) is a solution as well. d. If \(\vec{x}\) is a solution of the homogeneous system \(A \vec{x}=\overrightarrow{0}\) and \(k\) is an arbitrary constant, then \(k \vec{x}\) is a solution as well.

Short Answer

Expert verified
All homogeneous systems are consistent because at least the trivial solution exists (\(\vec{x}=\vec{0}\)). A homogeneous system with fewer equations than unknowns is underdetermined and thus has infinitely many solutions. The sum and scalar multiples of solutions to a homogeneous system are also solutions due to the properties of matrix multiplication and vector addition.

Step by step solution

01

Justifying that all homogeneous systems are consistent

A homogeneous system of equations can be represented as \(A\vec{x}=\vec{0}\), where \(A\) is a matrix, \(\vec{x}\) is the vector of variables, and \(\vec{0}\) is the zero vector. Since the zero vector is always in the column space of any matrix, there exists at least one solution, the trivial solution \(\vec{x}=\vec{0}\). Therefore, all homogeneous systems are consistent.
02

Demonstrating that a homogeneous system with fewer equations than unknowns has infinitely many solutions

If there are fewer equations than unknowns in a homogeneous system, it implies that the system is underdetermined. This means there are not enough equations to uniquely determine all the variables. As a result, there exist non-trivial solutions in addition to the trivial solution. Therefore, the system has infinitely many solutions, as these can be formed by combining the trivial and non-trivial solutions.
03

Showing that the sum of two solutions is also a solution

If \(\vec{x}_1\) and \(\vec{x}_2\) are solutions to the homogeneous system \(A\vec{x}=\vec{0}\), then by the properties of matrix multiplication, \(A(\vec{x}_1 + \vec{x}_2) = A\vec{x}_1 + A\vec{x}_2 = \vec{0} + \vec{0} = \vec{0}\). Hence, the vector \(\vec{x}_1 + \vec{x}_2\) is also a solution to the homogeneous system.
04

Proving that a scalar multiple of a solution is a solution

Let \(\vec{x}\) be a solution to the homogeneous system \(A\vec{x}=\vec{0}\), and let \(k\) be an arbitrary scalar constant. According to matrix multiplication rules, \(A(k\vec{x}) = k(A\vec{x}) = k\vec{0} = \vec{0}\). This demonstrates that \(k\vec{x}\) is also a solution to the homogeneous system.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trivial and Non-Trivial Solutions
In the study of homogeneous linear systems, where the equation is of the form \(A\vec{x} = \vec{0}\), understanding the concepts of trivial and non-trivial solutions is crucial. A trivial solution to the system is one where all the variable values are zero, which we denote as \(\vec{x} = \vec{0}\). Such a solution always exists because when you multiply any matrix with a zero vector, the result is a zero vector. Thus, homogeneous systems are said to be consistent, which means at least one solution exists. However, things get more interesting when we consider non-trivial solutions.

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