Find the linear approximation of the function
Use L(x) to approximate the numbers
In describing the solution of a system of linear equations with many solutions, why do we use a free variable as a parameter to describe the other variables in the solution? Why do we not we use a leading variable? Since by the commutative property of addition we can swap between the free and leading variables, e.g. x + y + z = x + z + y; The solution set will be essentially identical (albeit having different orders).
1. A parameter that is not a leading variable is referred to as a free variable.
2. A leading variable is the first variable in reduced form with a non-zero coefficient.
3. These definitions are clearest when applied to the Echelon form of a system of linear equations expressed as a Matrix.
Let S be the solution set of the system
Using the free variable z as the parameter
Using the leading variable y as the parameter
a) Determine a condition under which (x, y, z) is a linear combination of [-3, 5, -3], [-9, 11, -3], [-6, 8, -3]? Your condition should take the shape of a linear equation. I have the theorem that every vector (x, y, z) in is a linear combination of .