## Summer 2011: Question 3

*Suppose that is a positive integer. Pick intervals on the real number line. Assume that any pair of these intervals are disjoint. Pick real numbers .*

*Using ideas of linear algebra, prove that there is a polynomial function of degree at most**, with real number coefficients, so that**How many such polynomial functions are there? Justify your answer?*

*[HINT: Make a linear map. If the integral of a continuous function on an interval vanishes, then the function vanishes somewhere on the interval].*

### Solution

- [Vahid] Let and such that . That is, for each

*.*

This corresponds to equations in unknowns:

*.*

*.*

This is equivalent to the matrix equation , where is the vector , is the vector and is the matrix

.

A solution to this matrix equation gives a polynomial which satisfies the given condition.

## Summer 2010: Question 1

*Suppose that is a linear map. Suppose that are eigenvectors of with eigenvalues , and that these eigenvectors are all distinct. Prove that the eigenvectors are linearly independent.*

### Solution

We use induction on the number of eigenvectors . Let be the proposition that of the eigenvectors of are linearly independent.

is true because the set is linearly independent if — and eigenvectors are non-zero by definition.

Assume . That is assume if

for .

Consider . Multiply both sides of by :

.

By the inductive hypothesis () we have that for each . Since , it follows that in every case. Therefore we are left with

,

also.

Hence by the inductive hypothesis, the eigenvectors are linearly independent

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