We are tasked with showing that for an $m \times n$ matrix $A$ with $\text{rank}(A) = r$, the dimension of the solution space of $A \mathbf{x} = 0$ is $n - r$, without using the Rank-Nullity Theorem.

Solution Outline:

The equation we are solving is: $A \mathbf{x} = 0,$ where $A$ is an $m \times n$ matrix, and $\mathbf{x}$ is a vector in $\mathbb{R}^n$.

1. Definition of Rank:

The rank of a matrix $A$, denoted $r = \text{rank}(A)$, is the maximum number of linearly independent rows (or columns) in the matrix. This is the dimension of the column space of $A$, which is the subspace of $\mathbb{R}^m$ spanned by the columns of $A$.

2. Row-Reduced Echelon Form of $A$:

We perform elementary row operations on $A$ to bring it into row echelon form (REF). In this form, the rank of the matrix corresponds to the number of pivot columns (leading ones in the rows).

Let $A$ be row-reduced to a matrix $A'$ in row echelon form. This row echelon form has exactly $r$ pivot columns, each corresponding to a leading entry in a row, and $n - r$ non-pivot columns.

3. System of Equations:

The system $A \mathbf{x} = 0$ can be analyzed using the row echelon form $A'$. The equation $A' \mathbf{x} = 0$ represents a system of linear equations in $n$ variables. Since $A'$ has $r$ pivot columns, there are $r$ equations in the system.

• The non-pivot columns of $A'$ correspond to free variables in the system. These free variables will determine the solution space.
• The number of free variables in the system is $n - r$, since there are $n - r$ non-pivot columns.

4. Solution Space:

The solution space of the system $A \mathbf{x} = 0$ corresponds to the null space of $A$, which consists of all vectors $\mathbf{x} \in \mathbb{R}^n$ that satisfy the equation.

The number of free variables, $n - r$, directly tells us the number of dimensions in the solution space. Each free variable represents a degree of freedom in choosing the solution.

Conclusion:

Therefore, the dimension of the null space of $A$, or the solution space of $A \mathbf{x} = 0$, is $n - r$, as there are $n - r$ free variables.

Summary:

Without using the Rank-Nullity Theorem, we showed that the dimension of the solution space of $A \mathbf{x} = 0$ is $n - r$, by performing row reduction to echelon form and identifying the number of free variables, which equals the dimension of the null space.

Would you like further details or have any questions? Here are some related questions for you to explore:

1. What is the significance of pivot columns in the context of linear systems?
2. How do elementary row operations affect the solution space of a linear system?
3. Can we find the general solution to $A \mathbf{x} = 0$ by explicitly solving for the free variables?
4. How does the rank of a matrix relate to the linearly independent rows and columns?
5. What is the relationship between the rank of a matrix and its invertibility?

Tip: Always check whether you can reduce a matrix to row echelon form to better understand its solution space and determine the number of free variables.