Reduce A and B to echelon form, to find their ranks. Which...

Reduce $A$ and $B$ to echelon form, to find their ranks. Which variables are free? $$ A=\left[\begin{array}{llll} 1 & 2 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 1 & 2 & 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] $$ Find the special solutions to $A x=0$ and $B x=0 .$ Find all solutions.

Reduce $A$ and $B$ to echelon form, to find their ranks. Which variables are free?
$$
A=\left[\begin{array}{llll}
1 & 2 & 0 & 1 \\
0 & 1 & 1 & 0 \\
1 & 2 & 0 & 1
\end{array}\right] \quad B=\left[\begin{array}{lll}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{array}\right]
$$
Find the special solutions to $A x=0$ and $B x=0 .$ Find all solutions.

Key Concepts

Special Solutions

Special solutions refer to a set of particular solutions that form a basis for the null space of a matrix. They are derived by systematically assigning the free variables specific values (typically one free variable is set to 1 while the others are set to 0) to obtain independent solutions. Collectively, these special solutions allow reconstruction of all possible solutions to the homogeneous system by linear combination.

Free Variables

Free variables are variables in a system of linear equations that are not leading variables in the row echelon form; they can take on arbitrary values. Their presence signifies that the system has infinitely many solutions, and their assignments determine the structure of the solution set, particularly the null space of the matrix.

Null Space

The null space of a matrix is the set of all solutions to the homogeneous linear equation Ax=0. It describes the complete solution space for the system and its dimension is given by the number of free variables in the matrix after reducing it to echelon form. Understanding the null space is fundamental in linear algebra, as it reflects the inherent degrees of freedom in the system.

Row Echelon Form

Row echelon form is an arrangement of a matrix obtained by performing elementary row operations, where each nonzero row has its leading coefficient (pivot) to the right of the leading coefficient in the row above. This form simplifies solving systems of equations and clearly displays the independent rows, which assists in determining other properties like rank and free variables.

Matrix Rank

The rank of a matrix is the number of linearly independent rows or columns it possesses, typically determined by counting the nonzero rows in its row echelon form. The rank indicates the dimension of the row space and helps infer the number of free variables in a linear system, which is critical for solving homogeneous systems.

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