E is a vec torial space which has as
base B=(i^→ ,j^→ ,k^→ ). f: E→E is a linear
application such that
f(i^→ )=−i^→ +2k^→ ; f(j^→ )=j^→ +2k^→ and
j(k^→ )=2i^→ +2j^→ .
1. Write the matrix of f in base B.
2. Show that the kernel (ker f) of f
is a straigh line; give one base of its.
3.Determinate Im f.
we consider that application n≥1
det : M_n (R)→R
A det(A)
1−verify that ∀H∈M_n (R) and t∈R
if A=I_n ⇒det(A+tH)=1+t.Tr(H)+○(t)
2−suppose that: A∈GL_n (R)
prouve that the differntial of det in A is given by:
H Tr[(com(A))^T H]
3−determinate the differential of determinant of a matrix in general case.
(Use the density of GL_n (R) in M_n (R)
Tr: trace of matrix
(com(A))^T : transpose of the comatrix
Given a function f satisfies
f(x)= { ((x+a(√2) sin x ; 0≤x<(π/4))),((2x cot x +b ; (π/4)≤x≤(π/2))),((a cos 2x−bsin x ; (π/2)<x≤π)) :}
continuous in [ 0,π ], then find
the value of a and b.