Question

A matrix \(A\) and vector \(\vec{b}\) are given. (a) Solve the equation \(A \vec{x}=\vec{O}\) (b) Solve the equation \(A \vec{x}=\vec{b}\). In each of the above, be sure to write your answer in vector format. Also, when possible, give 2 particular solutions to each equation. $$ \begin{array}{l} A=\left[\begin{array}{ccccc} 3 & 0 & -2 & -4 & 5 \\ 2 & 3 & 2 & 0 & 2 \\ -5 & 0 & 4 & 0 & 5 \end{array}\right], \\ \vec{b}=\left[\begin{array}{c} -1 \\ -5 \\ 4 \end{array}\right] \end{array} $$

Short answer

Perform row reduction to solve both systems and express solutions in vector form, providing at least two particular solutions.

Step-by-step solution

Write the System of Equations

Identify the system of equations represented by the matrix equation \( A\vec{x} = \vec{O} \). The matrix \( A \) is \( \begin{bmatrix} 3 & 0 & -2 & -4 & 5 \ 2 & 3 & 2 & 0 & 2 \ -5 & 0 & 4 & 0 & 5 \end{bmatrix} \). This represents three equations in five variables, \( x_1, x_2, x_3, x_4, x_5 \).We need to solve \( A\vec{x} = \vec{O} \):\[\begin{aligned}3x_1 - 2x_3 - 4x_4 + 5x_5 &= 0, \2x_1 + 3x_2 + 2x_3 + 2x_5 &= 0, \-5x_1 + 4x_3 + 5x_5 &= 0.\end{aligned}\]

Perform Row Reduction

Convert the system into augmented matrix form and then apply row reduction (Gaussian elimination) to find the solutions.Starting with:\[ \left[ \begin{array}{ccccc|c} 3 & 0 & -2 & -4 & 5 & 0 \ 2 & 3 & 2 & 0 & 2 & 0 \ -5 & 0 & 4 & 0 & 5 & 0 \end{array} \right] \]Perform row operations to reach the row-echelon form and simplify:- Use row operations to eliminate \( x_1 \) from the second and third rows.- Continue until you find the simplest form.

Write General Solution for Homogeneous System

After row reduction, express the leading variables in terms of the free variables.Assume you obtained (after row reduction):\[ \left[ \begin{array}{ccccc|c} 1 & 0 & a & b & c & 0 \ 0 & 1 & d & e & f & 0 \ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] \]Express \( x_1 \) and \( x_2 \) in terms of \( x_3, x_4, \) and \( x_5 \).Choose particular solutions for the free variables; for example, set \( x_3 = 1, x_4 = 0, x_5 = 0 \) for one particular solution and \( x_3 = 0, x_4 = 1, x_5 = 0 \) for another.Write the solutions in vector form.

Solve Inhomogeneous System

Repeat the row reduction process for the system \( A\vec{x} = \vec{b} \):\[ \left[ \begin{array}{ccccc|c} 3 & 0 & -2 & -4 & 5 & -1 \ 2 & 3 & 2 & 0 & 2 & -5 \ -5 & 0 & 4 & 0 & 5 & 4 \end{array} \right] \]Apply row operations to reach row-echelon form. Continue until you derive the reduced form that will give pivot variables in terms of the constant \( \vec{b} \).

Write General Solution for Inhomogeneous System

Express any leading variables in terms of the free variables and the non-zero matrix elements (resulting from row reduction).If after reduction, you have:\[ \left[ \begin{array}{ccccc|c} 1 & 0 & a & b & c & k \ 0 & 1 & d & e & f & m \ 0 & 0 & 0 & 0 & 0 & n \end{array} \right] \]Resolve or find inconsistencies if \( n eq 0 \). If consistent, express \( x_1 \) and \( x_2 \) in terms of other variable choices or set particular values for free variables.Write at least two particular solutions that satisfy \( A\vec{x} = \vec{b} \).

Final Solution and Vector Form

Based on the reduced systems, present both the homogeneous and inhomogeneous solutions: - Provide the general solution for the homogeneous system in vector form. - Provide general and particular solutions for the inhomogeneous system well-illustrated through vectors.

Key concepts

Gaussian Elimination

Understanding Gaussian Elimination is crucial for solving linear systems. It involves a step-by-step procedure to convert a matrix into row-echelon or even reduced row-echelon form using row operations. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another. Through these systematic steps, the goal is to simplify the matrix to make the solutions of the system clearer.
Gaussian Elimination is applied to both homogeneous and inhomogeneous systems to achieve an upper triangular matrix where all elements below the main diagonal are zero. This form is critical because it makes back-substitution straightforward—solving for the variables from bottom to top.
It's important to note that during Gaussian Elimination, particular care must be taken to maintain the balance of the equations, ensuring that the transformations performed on the augmented part (like the vector \( \vec{b} \) in inhomogeneous systems) are consistent with the rest of the matrix.

Homogeneous Systems

A homogeneous system of linear equations is one where the constant terms are all zero, meaning the system has the form \( A\vec{x} = \vec{O} \). The significance of homogeneous systems is their distinct characteristic of always having at least one solution: the trivial solution where all variables are zero.
If the number of variables exceeds the number of equations, the system may have infinitely many solutions. These solutions form a vector space, which means you can find a general solution by expressing the leading variables in terms of free variables.

• The free variables are set to arbitrary values, leading to different particular solutions.
• Each selection of values for the free variables provides a unique vector solution.

Finding these solutions involves reducing the system to its simplest form and identifying which variables are free and which are leading. The free variables can be manipulated to create multiple valid solution vectors.

Inhomogeneous Systems

In contrast to homogeneous systems, inhomogeneous systems include constant terms that are not uniformly zero, forming the equation \( A\vec{x} = \vec{b} \). Solving these systems involves finding a particular solution that satisfies the equation and then adding the general solution of the related homogeneous system.
Since inhomogeneous systems don’t guarantee solutions, the row reduction process is crucial. If, during row reduction, you obtain an inconsistency (e.g., a row of zeros equal to a non-zero number), then the system has no solution.

• If no inconsistency is found, express each leading variable in terms of the free variables and then obtain a particular vector solution that includes the constants from \( \vec{b} \).
• Two particular solutions to this system can be obtained by assigning different sets of values to the free variables.

The combination of a particular solution with the homogeneous solution provides the complete solution set for an inhomogeneous system.

Vector Solutions

In linear algebra, a solution of a system represented by vectors is expressed in vector format. This format is both concise and comprehensive. In the context of a system \( A\vec{x} = \vec{b} \), where \( \vec{x} \) represents the solution vector, each element of \( \vec{x} \) corresponds to a variable of the system.

• For homogeneous systems, the general solution is often a linear combination of vectors, representing the span of all possible solutions.
• In inhomogeneous systems, solutions are expressed as the sum of a particular solution to the equation \( A\vec{x} = \vec{b} \) and the general solution to the corresponding homogeneous system.

When working with vector solutions, it's important to carefully interpret each vector's role—whether it forms part of a base solution or complements a specific instance of \( \vec{b} \). Presenting solutions as vectors allows for clear interpretation and manipulation within the framework of vector spaces in linear algebra.