(a) Show that 𝐱1^⋆=( 1 1 0 ) and 𝐱2^⋆=( -3 3 -2 ) are particular

step 1 so the question says that find general solution of given system so we have determinant of a minu

(a) Show that 𝐱1^⋆=( 1 1 0 ) and 𝐱2^⋆=( ...

(a) Show that $\mathbf{x}_1^{\star}=\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)$ and $\mathbf{x}_2^{\star}=\left(\begin{array}{r}-3 \\ 3 \\ -2\end{array}\right)$ are particular solutions to the system $\left(\begin{array}{rrr}2 & -1 & -5 \\ 1 & -4 & -6 \\ 3 & 2 & -4\end{array}\right) \mathbf{x}=\left(\begin{array}{r}1 \\ -3 \\ 5\end{array}\right)$. (b) Find the general solution.

Key Concepts

Null Space of a Matrix

The null space (or kernel) of a matrix is the set of all vectors that, when multiplied by the matrix, result in the zero vector. In the context of linear systems, the null space represents the solution set of the homogeneous equation, and it determines the structure of the general solution by indicating the degrees of freedom or free parameters present in the system.

General Solution

The general solution of a linear system encompasses all possible solutions. It is typically expressed as the sum of a particular solution and the general solution of the associated homogeneous system. This representation captures both a specific instance that satisfies the system and the directions in which additional solutions can be generated.

Particular Solutions

A particular solution to a linear system is a specific set of values for the variables that satisfies the system of equations. It is used as a starting point to understand the structure of the solution set, and even if the system has infinitely many solutions, any one specific solution can serve as a particular solution.

Systems of Linear Equations

A system of linear equations involves finding values for variables that simultaneously satisfy multiple linear equations. These systems are typically represented in matrix form, which allows for efficient methods of verification and computation through techniques such as substitution, elimination, and matrix operations such as inversion or row reduction.

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