Discovering The Particular Solution Space Of Ax=B With Homogeneous Solutions

the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax = 0

suppose the equation Ax = b is consistent for some given b, and let p be a solution. Then the solution set of AX = b is …

Let A be a matrix, and consider the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax = 0.

This set of vectors is known as the particular solution space of Ax=b, where b is any fixed vector.

To understand why this is the case, let’s first consider what it means for vh to be a solution of Ax=0. This means that when we multiply A by vh, the result is the zero vector. In other words, A(vh) = 0.

Now, suppose that we have a specific vector p and we want to find a solution w of the equation Ax = b. We can write b as the sum of two vectors: b = x + y, where x is any solution of Ax = b (i.e. the homogeneous solution) and y is the particular solution (i.e. the non-homogeneous part).

Now, let’s consider the vector w = p + vh. We can calculate Ax as follows:

Ax = A(p+vh) = Ap + Avh

But we know that Avh = 0, since vh is a solution of Ax=0. Therefore, we have:

Ax = Ap

So, in order for w = p + vh to be a solution of Ax = b, we need:

Ap = x + y

or equivalently:

y = Ap – x

In other words, y is the difference between the matrix A applied to p and the homogeneous solution x. This means that any particular solution of Ax = b can be written as the sum of a specific solution p and a particular solution of the homogeneous equation Ax = 0.

Thus, the set of all vectors of the form w = p + vh, where vh is any solution of the homogeneous equation Ax = 0, forms the particular solution space of Ax=b.

More Answers: