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SpiffyEh
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Homework Statement
Suppose you have a set S of three points in R2,
S = {(t1, p1), (t2, p2), (t3, p3)} ;
which you seek to interpolate with the quadratic polynomial p(t) = a0 + a1t + a2t2.
5.1. Interpolations and Linear Systems. Using S and p(t) define a linear system of equations in the a0; a1; a2 variables.
5.2. Existence and Uniqueness in Interpolation. Determine which of the following sets of points can be uniquely interpolated by p(t).
S1 = {(1, 12), (2, 15), (3, 16)}
S2 = {(1, 12), (1, 15), (3, 16)}
S3 = {(1, 12), (2, 15), (2, 15)}
5.1 - Hint: This problem is meant to trick you. Clearly, p(t) is a nonlinear equation in the t-variable but once you have chosen a t-value then it is a
linear equation in the coecient variables. If you choose many t-values then you have many linear equations and now the tools of linear algebra apply.
5.2 - Your choice! We have two ways to approach this problem. First, you have a linear system and thus row-reduction and interpretation of pivot
structure. However, if you think about the geometry of the points and the possible graphs of quadratic polynomials you should be able to determine,
which of the points can be interpolated.
I have no idea what to do or where to start. We never talked about interpolation. Can someone explain how to do this to me so that I can actually try it?
Suppose you have a set S of three points in R2,
S = {(t1, p1), (t2, p2), (t3, p3)} ;
which you seek to interpolate with the quadratic polynomial p(t) = a0 + a1t + a2t2.
5.1. Interpolations and Linear Systems. Using S and p(t) define a linear system of equations in the a0; a1; a2 variables.
5.2. Existence and Uniqueness in Interpolation. Determine which of the following sets of points can be uniquely interpolated by p(t).
S1 = {(1, 12), (2, 15), (3, 16)}
S2 = {(1, 12), (1, 15), (3, 16)}
S3 = {(1, 12), (2, 15), (2, 15)}
Homework Equations
5.1 - Hint: This problem is meant to trick you. Clearly, p(t) is a nonlinear equation in the t-variable but once you have chosen a t-value then it is a
linear equation in the coecient variables. If you choose many t-values then you have many linear equations and now the tools of linear algebra apply.
5.2 - Your choice! We have two ways to approach this problem. First, you have a linear system and thus row-reduction and interpretation of pivot
structure. However, if you think about the geometry of the points and the possible graphs of quadratic polynomials you should be able to determine,
which of the points can be interpolated.
The Attempt at a Solution
I have no idea what to do or where to start. We never talked about interpolation. Can someone explain how to do this to me so that I can actually try it?