Find all solutions to the homogeneous system A 𝐱=0 for the coefficient matrix (a) ( 3 -1

step 1 So we are required to solve the homogeneous linear system corresponding to the given coefficient

Find all solutions to the homogeneous system A 𝐱=0 for the...

Find all solutions to the homogeneous system $A \mathbf{x}=\mathbf{0}$ for the coefficient matrix (a) $\left(\begin{array}{rr}3 & -1 \\ -9 & 3\end{array}\right)$, (b) $\left(\begin{array}{rrr}2 & -1 & 4 \\ 3 & 1 & 2\end{array}\right)$, (c) $\left(\begin{array}{rrrr}1 & -2 & 3 & -3 \\ 2 & 1 & 4 & 0\end{array}\right)$, (d) $\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right)$, (e) $\left(\begin{array}{rrr}0 & 2 & -1 \\ -2 & 0 & 3 \\ 1 & 3 & 0\end{array}\right)$, (f) $\left(\begin{array}{rr}1 & -2 \\ 1 & -1 \\ 2 & -1 \\ 1 & 0\end{array}\right)$, (g) $\left(\begin{array}{rrr}1 & 2 & 0 \\ -1 & -3 & 2 \\ 4 & 7 & 2 \\ -1 & 1 & 6\end{array}\right)$, (h) $\left(\begin{array}{rrrr}0 & 0 & 3 & -3 \\ 1 & -1 & 0 & 3 \\ 2 & -2 & 1 & 5 \\ -1 & 1 & 1 & -4\end{array}\right)$.

Find all solutions to the homogeneous system $A \mathbf{x}=\mathbf{0}$ for the coefficient matrix
(a) $\left(\begin{array}{rr}3 & -1 \\ -9 & 3\end{array}\right)$,
(b) $\left(\begin{array}{rrr}2 & -1 & 4 \\ 3 & 1 & 2\end{array}\right)$,
(c) $\left(\begin{array}{rrrr}1 & -2 & 3 & -3 \\ 2 & 1 & 4 & 0\end{array}\right)$,
(d) $\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right)$,
(e) $\left(\begin{array}{rrr}0 & 2 & -1 \\ -2 & 0 & 3 \\ 1 & 3 & 0\end{array}\right)$,
(f) $\left(\begin{array}{rr}1 & -2 \\ 1 & -1 \\ 2 & -1 \\ 1 & 0\end{array}\right)$,
(g) $\left(\begin{array}{rrr}1 & 2 & 0 \\ -1 & -3 & 2 \\ 4 & 7 & 2 \\ -1 & 1 & 6\end{array}\right)$,
(h) $\left(\begin{array}{rrrr}0 & 0 & 3 & -3 \\ 1 & -1 & 0 & 3 \\ 2 & -2 & 1 & 5 \\ -1 & 1 & 1 & -4\end{array}\right)$.

Key Concepts

Homogeneous Systems

A homogeneous system is a system of linear equations in which all of the constant terms are zero. This means the system can be written in the form A·x = 0. Such systems always have at least one solution (the trivial solution, where all variables are zero), and in many cases they possess additional, nontrivial solutions that form a subspace of the vector space.

Null Space

The null space of a matrix A is the set of all vectors x that satisfy A·x = 0. This subspace is central to understanding the solutions of homogeneous systems. The null space shows the degrees of freedom in the system and its dimension, known as the nullity, indicates how many parameters are needed to describe the complete set of solutions.

Gaussian Elimination

Gaussian elimination is a method used to simplify a system of linear equations through a series of row operations. By transforming the matrix into an echelon form or a reduced row echelon form, one can easily identify the pivot positions and free variables. This systematic approach simplifies the analysis of the solution set for the homogeneous system.

Free and Pivot Variables

In the context of solving linear systems, the pivot variables are those corresponding to leading ones in the row-reduced matrix, while the free variables are not associated with a pivot. Free variables can be assigned arbitrary values, and pivot variables are then expressed in terms of these free parameters. This distinction is crucial for expressing the general solution of a homogeneous system.

Parametric Vector Form

Once the system is reduced and free variables are identified, the general solution can be expressed in parametric vector form. This involves writing the solution as a linear combination of vectors, where the free variables act as parameters. The resulting expression clearly shows the structure of the solution space, including its dimension and spanning set.

Rank-Nullity Theorem

The Rank-Nullity Theorem is a fundamental result in linear algebra when studying systems like A·x = 0. It states that the sum of the rank of the matrix (the number of independent rows or columns) and the nullity (the dimension of the null space) equals the total number of columns of the matrix. This theorem provides insight into the relationship between the constraints posed by the equations and the freedom available in the solution space.

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