Determine if the following statement is true or false:The number of free variables in the equationAx=0equals the dimension of Nul(A).A. TrueB. False

To determine if the statement is true or false, we need to understand the relationship between the number of free variables in the equation Ax=0 and the dimension of Nul(A).

1. First, let's define the terms:

   • Ax=0 represents a homogeneous system of linear equations, where A is a matrix and x is a vector of variables.
   • Nul(A) represents the null space of matrix A, which is the set of all solutions to the equation Ax=0.

2. The number of free variables in the equation Ax=0 corresponds to the number of variables that can take any value without affecting the solution. These variables are typically represented by parameters.

3. The dimension of Nul(A) represents the number of linearly independent vectors that span the null space of matrix A.

4. Now, let's analyze the relationship between the number of free variables and the dimension of Nul(A):

   • If the number of free variables in the equation Ax=0 is equal to the dimension of Nul(A), then the statement is true.
   • If the number of free variables in the equation Ax=0 is not equal to the dimension of Nul(A), then the statement is false.

5. Therefore, to determine if the statement is true or false, we need to compare the number of free variables in the equation Ax=0 with the dimension of Nul(A).