A linear system is a set of equations that can be represented in the form Ax = y, where A is a matrix of coefficients, x is a vector of variables, and y is a vector of constants. The goal of solving a linear system is to find values for the variables that satisfy all of the equations.
In the context of a linear system, "Ax = y" represents the matrix equation that needs to be solved. A is the coefficient matrix, x is the variable vector, and y is the constant vector. This equation can also be written as a system of equations, where each row of A and the corresponding element in x are multiplied together and summed to equal the element in y.
In order to solve a linear system in \mathbb{R}^3 (three-dimensional space), you will need to use three equations with three variables. These equations can be represented in the form Ax = y, where A is a 3x3 matrix, x is a vector with three elements, and y is a vector with three elements. You can then use various methods, such as Gaussian elimination or matrix inversion, to solve for the values of x that satisfy all three equations.
Yes, a linear system can have multiple solutions. In fact, it often does. This means that there are multiple combinations of values for the variables that satisfy all of the equations in the system. These solutions can be represented as points in \mathbb{R}^3 that lie on the intersection of the planes represented by the equations in the system.
Solving "Ax = z" is not fundamentally different from solving "Ax = y". Both equations represent linear systems and can be solved using the same methods. The only difference is that the constant vector, z, may have different values than the constant vector, y. This will result in different solutions for the variable vector, x.