1.3 Applications of Systems of Linear Equations

Question 1

Generally, the farther points are from those used to fit a polynomial ________

Answer 1

the worse the fit

Question 2

With n points in an xy-plane, you can use a system of linear equations to find a polynomial function of degree ______. This procedure is called _______.

Answer 2

n - 1 ; polynomial curve fitting

Question 3

Describe the process to solve for the n coefficients for a polynomial function p(x)

Answer 3

1. Choose a polynomial of one degree less than the number of points
2. Substitute each points into one iteration of the selected polynomial function to produce a system of linear equations
3. Solve for the system of linear equations. These solutions will be the coefficients in the polynomial function.

Question 4

Provide an example of when translating large x values would be useful before curve fitting

Answer 4

For data involving dates, it can be useful for the x value to be translated into deviation from a given year. (e.g. z = x - 2013)

Question 5

(T/F) A system of linear equations with fewer equations than variables always has at least one solution

Answer 5

False

Question 6

Underdetermined

Answer 6

More variables than equations

Question 7

Overdetermined

Answer 7

More equations than variables

Question 8

How many solutions can an underdetermined system have?

Answer 8

The underdetermined system can have infinitely many or no solutions, but never a unique solution.

Question 9

How many solutions can an overdetermined system have?

Answer 9

The overdetermined system can have infinitely many, no solutions, or a unique solution. This is just like a balanced system with the same number of equations and variables