If V is a vector space B = {e_{1}, ..., e_{d}} be a basis of V and T : V → V is a linear transformation we denote by A_{B}(T) the matrix of T in the basis B. That is A_{B}(T) = (a_{j}^{i}), 1 ≤ i, j ≤ d is a matrix such that T (e_{j}) = ∑_{i=1}^{d} a_{j}^{i}e_{i}.

1. a)Let B_{0} =: {e_{1}, e_{2}} be the standard basis of ℝ^{2}, T : ℝ^{2} → ℝ^{2} be a linear transformation such that

A_{B_{0}}(T) = (\begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} )

[that is

T (e_{1}) = 2e_{1} + e_{2}, T (e_{2}) = e_{1} + e_{2}]

Find a basis B =: {f_{1}, f_{2}} of ℝ^{2} such that the matrix A_{B}(T) is

diagonal. That is A_{B}(T) = Λ = (\begin{matrix}λ_{1} & 0 \\ 0 & λ_{2} \end{matrix} )

b)find the numbers λ_{1} and λ_{2}. c)Let f_{1} = c_{1}^{1}e_{1} + c_{1}^{2}e_{2}, f_{2} = c_{2}^{1}e_{1} + c_{2}^{2}e_{2}, c_{j}^{i} ∈ ℝ. Show that A = CΛC^{−1} where A := A_{B_{0}}(T).

C = (\begin{matrix}c_{1}^{1} & c_{1}^{2} \\ c_{2}^{1} & c_{2}^{2} \end{matrix} )

d)Let a = (3 − \sqrt{5})/2. Find the elements of the matrix a^{1000}A^{1000} with the precision of 10 percent. ”With the precision of 10 percent” means that if e is the exact (unknown) answer and a is the approximate answer which you give, you must have |a − e|/a < 0.1.

2.a) Let f be a differentiable function on ℝ such that f′(x) ≡ 1. Prove that there exists c ∈ ℝ such that f(x) = x + c for all x ∈ ℝ. b)Find all twice differetiable functions F on ℝ such that F′′(x) ≡ 1.

Let f : Mat(n, n) → Mat(n, n) be a function such that f(A) = A^{2}.

3. a)Show that the function f is differentiable and for any n×n matrix H find the differential df(H) : Mat(n, n) → Mat(n, n).

We can identify the vector space Mat(n, n) with ℝ^{n^{2}}, A → {a_{i,j}}. Therefore we can define the partial derivatives ∂f/∂a_{i,j}.

b) Find the partial derivatives ∂f/∂a_{i,j}.

c) Find the differential dF (H) : Mat(n, n) → Mat(n, n) where F (A) := A^{3}.

4. a) Let f be a differentiable function on ℝ and F be and function on ℝ^{2} defined by F (x, y) := x^{7}f(y/x^{2}).

Show that x∂F/∂x + 2y∂F/∂y = 7F

b) Find a generalization of this result.

If f is an infinitely differetiable function on ℝ at a ∈ ℝ we can define the Taylor series

P_{f}(a, x) := ∑_{n≥0} c_{n}(x − a)^{n}, c_{n} := f^{(n)}(a)/n!

. We define by I_{a} ⊂ [0, ∞) as the set of numbers r ≥ 0 such that c_{n}r^{n} → 0 for n → ∞ and write r_{f}(a) := lub(I_{a}). [It could be that r_{f}(a) = ∞].

Let f be a function on ℝ − \{0, 1, 2\} given by f(x) := \frac{1}{x(x−1)(x−2)}.

5. a)Show that f is an infinitely differetiable function on ℝ−{0, 1, 2} b)Find r_{f}(a) for any a ∈ ℝ, a ≠ 0, 1, 2

Let V be a vector space, Q(V) the space of quadratic functions on V and B_{s}^{2}(V) be the space of symmetric biliner forms on V. To any symmetric biliner form f on V we associate a function q(f) on V where q(f)(v) := f(v, v).

6. a) Show that for any f ∈ B_{s}^{2}(V) the function q(f) on V is quadratic.

Part a) shows that we have defined a map α : B_{s}^{2}(V) → Q(V), α(f) := q(f).

b)Construct explicitely a linear map β : Q(V) → B_{s}^{2}(V) which is the inverse of α. (Note that it follows from this that α gives an isomorphism between the vector spaces B_{s}^{2}(V) and Q(V).)

c)Suggest a generalization of of the construction f → q(f) and try to generalize the problems a)-b).

7. Let f be a twice differentiable function on ℝ^{2} and g : ℝ^{2}−{x, 0} → ℝ^{2} be given by g(x, y) := (xy, x/y).

Find first and second partial derivatives of the function F := f ∘ g in terms of the partial derivatives of the function f