MATH 395 PROBLEMS 3

IGOR KRIZ

Regular problems:

1. Find the eigenvalues of

(a) ( \begin{matrix}1 & 2 \\ 3 & 2 \end{matrix} )

(b) ( \begin{matrix}7 & 0 & −4 \\ 0 & 5 & 0 \\ 5 & 0 & −2 \end{matrix} ).

2. Using the geometric interpretation of determinants, find the volume of the solid in ℝ^{3} consisting of all (x, y, z) such that

(2x + 3y + z)^{2} + (x − 2y + 3z)^{2} + (x + y + z)^{2} ≤ 6.

[Find a linear transformation which maps the solid onto a ball, and consider the determinant of that transformation.]

3. For x, y ∈ ℝ, define

d_{1}(x, y) = (x − y)^{2},
d_{2}(x, y) = \sqrt{|x − y|},
d_{3}(x, y) = |x^{2} − y^{2}|,
d_{4}(x, y) = |x − 2y|,
d_{5}(x, y) = \frac{|x−y|}{1+|x−y|}.

Determine, for each of these, whether it is a metric or not. [Recall that a metric is supposed to satisfy (1) d(x, y) = 0 if and only if x = y, (2) d(x, y) = d(y, x) and (3) d(x, z) ≤ d(x, y) + d(y, z).]

4. Prove that if an invertible matrix is upper triangular (a_{ij} = 0 for i > j), then the inverse is also upper triangular.

Challenge problems:

5. Prove the following fact: Let n data pairs (x_{1}, y_{1}),... , (x_{n}, y_{n}) be given such that the x_{i} are distinct. Then there exists a unique polynomial

p(x) = r_{0} + r_{1}x + ... + r_{n−1}x^{n−1}

such that p(x_{i}) = y_{i}. [Hint: write a system of linear equations for the r_{i}’s, solve using Cramer’s rule and determinants.]

6. Let X be a metric space. Let X̄ be the set of equivalence classes of all Cauchy sequences in X under the following equivalence relation: Cauchy sequences (a_{n}) and (b_{n}) are equivalent if the sequence (a_{1}, b_{1}, a_{2}, b_{2}, ...) is Cauchy (prove that this is an equivalence relation). Now if the metric on X is d, on X̄, we can define

d((a_{n}), (b_{n})) = lim_{n→∞} d(a_{n}, b_{n}).

(a) Prove that this makes X̄ into a metric space.

(b) Prove that the map X → X̄ given by a ↦ (a, a, ...) preserves metric (=is an isometry), and is an embedding.

(c) Prove that X̄ is a complete metric space (i.e. every Cauchy sequence in X̄ converges).