MATH 296 PROBLEMS 7

IGOR KRIZ

Regular problems:

1. Find the base change matrix from the basis

( \begin{matrix}2 \\ 3 \\ 1 \end{matrix} ) , ( \begin{matrix}3 \\ 5 \\ 1 \end{matrix} ) , ( \begin{matrix}2 \\ 2 \\ 2 \end{matrix} )

to the basis

( \begin{matrix}1 \\ 1 \\ 1 \end{matrix} ) , ( \begin{matrix}1 \\ 2 \\ 3 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \\ 1 \end{matrix} ) .

[Use the definition.]

2. Recall that P_{n} denotes the vector space of polynomials of degrees ≤ n. Prove that the derivative is a linear map P_{n} → P_{n}. Find the matrix of this map with respect to the basis {1, x, x^{2}, ..., x^{n}} in the source and the same basis in the target.

3. Prove that the identity Id : ℝ^{n} → ℝ^{n} is a homeomorphism of metric spaces where the metric is the Euclidean metric in the source and the Postman metric ρ(x, y) = ∑ |x_{i} − y_{i}| in the target.

4. Find a basis of the subspace of ℝ^{4} consisting of all ( \begin{matrix}x \\ y \\ z \\ t \end{matrix} ) such that

x + 2y + 3z − t = 0.

[It is a null space.] Challenge problems:

5. By ℝ_{n} we denote the vector space of all 1×n matrices (row vectors). Now let A be an m × n matrix. The row space of A is the subspace of ℝ_{n} 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 f : X → Y is continuous if and only if for every open subset U ⊆ Y, f^{−1}(U) is open in X (U ⊆ Y is open if for every x ∈ Y, there exists an ϵ > 0 such that U_{ϵ}(x) ⊆ U).