Translating The Solution Set Of Homogeneous Systems: The Role Of Particular Solutions

translating the solution set of Ax = 0, using any particular solution p of Ax = b for the translation

Theorem 6 says that if Ax = b has a solution, then the solution set is obtained by …

To translate the solution set of Ax = 0, using any particular solution p of Ax = b, we first need to understand the concept of a homogeneous system and its associated solution set.

A homogeneous system of linear equations is one where all the constants on the right-hand side of the equations are equal to zero. In other words, it is a system of equations Ax = 0, where x is a vector of unknowns and A is a matrix of coefficients. The solution set of Ax=0 is the set of all possible values of x that satisfy the system of equations.

Now, suppose we have another system of linear equations Ax = b, where b is a non-zero vector of constants. If we can find a particular solution p of Ax = b, then we can use this solution to translate the solution set of the homogeneous system Ax = 0.

Specifically, let q be any solution of Ax = 0. Now consider the vector p + q, which is a solution to Ax=b. Why? Because A(p+q) = Ap + Aq = b + 0 = b, since Ap=b by assumption and Aq=0 by the fact that q is a solution to Ax=0.

So we’ve shown that p+q is a solution to Ax=b. Moreover, we know that q is a solution to Ax=0, so we have p+q – p = q is also a solution to Ax=b.

Therefore, we’ve shown that any solution q of Ax = 0 can be expressed as q = p + x, where x is a solution to Ax = 0. This means that the solution set of Ax = 0 can be obtained by adding p to every vector in the solution set of Ax = b.

In summary, we can translate the solution set of Ax = 0 by adding any particular solution p of Ax = b to every vector in the solution set of Ax = 0.

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