Question

difference between homogeneous and nonhomogeneous equation is Equal Tosine cosine source​

Answer 1

Answer:

A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. ... A nonhomogeneous system has an associated homogeneous system, which you get by replacing the constant term in each equation with zero.

Answer 2

Ans: A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. When a row operation is applied to a homogeneous system, the new system is still homogeneous. It is important to note that when we represent a homogeneous system as a matrix, we often leave off the final column of constant terms, since applying row operations would not modify that column. So, we use a regular matrix instead of an augmented matrix. Of course, when looking for a solution, it's important to take the constant zero terms into account.

A nonhomogeneous system has an associated homogeneous system, which you get by replacing the constant term in each equation with zero. Section 1.I.3 in the textbook is about understanding the structure of solution sets of homogeneous and non-homogeneous systems. The main theorems that are proved in this section are:

Theorem: The solution set of a homogeneous linear system with n variables is of the form, where k is the number of free variables in an echelon form of the system and are [constant] vectors in Rn.

Theorem: Consider a system of linear equations in n variables, and suppose that p is a solution of the system. Then the solution set of the system is of the form