Page 519 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

(a) Identify the bases for used for interpolation in the standard form (found by using the Vandermonde system), the

Newton form, and the Lagrange form, assuming , , and .

(b) What is the transition matrix from the Newton form basis to the standard basis?

To write the coordinate vector for a vector, it is necessary to specify an order for the vectors in the basis. If P is the transition
14. matrix from a basis to a basis B, what is the effect on P if we reverse the order of vectors in B from , …, to , …,

? What is the effect on P if we reverse the order of vectors in both and B?

Consider the matrix
15.

(a) P is the transition matrix from what basis B to the standard basis for ?

(b) P is the transition matrix from the standard basis to what basis B for ?

The matrix
16.

is the transition matrix from what basis B to the basis {(1, 1, 1), (1, 1, 0), (1, 0, 0)} for ?

If holds for all vectors w in , what can you say about the basis B?
17.

Indicate whether each statement is always true or sometimes false. Justify your answer by giving a
18. logical argument or a counterexample.

(a) Given two bases for the same inner product space, there is always a transition matrix from
one basis to the other basis.

(b) The transition matrix from B to B is always the identify matrix.

(c) Any invertible matrix is the transition matrix for some pair of bases for .