- #1
mattmns
- 1,128
- 6
Just a quick question. If I have a system of linear equations, and I find a basis for the space of solutions, then each element in the basis should be a solution to the system right? Thanks!
You are right about a, that was my typo0rthodontist said:Well, for a the correct answer is (1, -2, 0). For c the dimension is actually 2.
The basis of the space of solutions of a linear system is a set of vectors that can be used to represent all possible solutions to the system. It is a fundamental concept in linear algebra and is used to understand the structure of the solution space.
The basis of the space of solutions can be found by first solving the linear system using techniques such as Gaussian elimination or matrix inversion. Then, the columns of the resulting solution matrix that contain the leading variables form the basis of the solution space.
Yes, the basis of the space of solutions can be unique. This occurs when the linear system has a unique solution or when the solution space is one-dimensional, meaning that there is only one linearly independent vector that can represent all solutions.
The dimension of the basis of the space of solutions is equal to the number of variables in the linear system minus the rank of the coefficient matrix. This means that the dimension of the basis can vary depending on the number of variables and the structure of the linear system.
Yes, the basis of the space of solutions can change if the linear system is transformed. This can happen if the transformation affects the rank of the coefficient matrix or if the transformation introduces new variables to the system, changing the dimension of the solution space.