**Igor Kriz**

**Regular problems:**

**1.**

Find the base change matrix from the basis

to the basis

[Use the definition.]

**2.**

Recall that denotes the vector space of polynomials of degrees . Prove that the derivative is a linear map . Find the matrix of this map with respect to the basis in the source and the same basis in the target.

**3.**

Prove that the identity is a homeomorphism of metric spaces where the metric is the Euclidean metric in the source and the Postman metric in the target.

**4.**

Find a basis of the subspace of consisting of all such that

[It is a null space.]

**Challenge problems:**

**5.**

By we denote the vector space of all
matrices (row vectors). Now let *A* be an matrix.
The *row space* of *A* is the subspace of
spanned by the rows of *A*. Prove that the dimension of the column
space of *A* (called the *column rank*) is equal to the dimension
of the row space of *A* (called the *row rank*).
[We already know that equivalent row operations preserve the dimension
of the column space. Show that equivalent row operations actually
preserve the row space. Thus, it suffices to verify the statement for
matrices in reduced row echelon form.]

**6.**

Prove that a map of metric spaces is continuous if
and only if for every open subset , is open
in *X*
( is open if for every , there exists an
such that ).

Sun Feb 22 23:31:30 EST 1998