Understanding the Difference Between Solution Sets: Ax = b vs Ax = 0

The solution set of Ax = b is always obtained by translating the solution set of Ax = 0.

The given statement is not correct

The given statement is not correct. The solution set of Ax = b is not always obtained by translating the solution set of Ax = 0.

To understand why, let’s consider a system of linear equations in matrix form: Ax = b. Here, A is the coefficient matrix, x is the vector of variables, and b is the constant vector. The solution set of this system represents the values of x that satisfy all the equations simultaneously.

When we have Ax = 0, we are considering the homogeneous system, where the constant vector b is zero. The solution set of Ax = 0 represents the set of all vectors that satisfy the homogeneous system. It is also referred to as the null space or kernel of the matrix A.

On the other hand, when we have Ax = b, where b is a non-zero vector, the solution set represents the set of all vectors that satisfy the non-homogeneous system. This solution set will generally contain both the null space of A (solutions to Ax = 0) as well as additional vectors that satisfy the non-homogeneous term.

In other words, the solution set of Ax = b is not just a translation of the solution set of Ax = 0, but rather a combination of the null space solutions and specific solutions that satisfy the non-homogeneous term b.

To summarize, while translating the solution set of Ax = 0 by adding a specific vector might give one solution to Ax = b, it does not provide the complete solution set. The solution set of Ax = b is obtained by considering both the null space solutions and specific solutions satisfying the non-homogeneous term b.

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