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Linear Algebra
1.)Use De Moivre’s Theorem to
a.)derive the 4th roots of w =-8i
b.) express cos(4θ) and sin(5θ) in terms of powers of cos θ and sin θ
c.) expand cos^6 θ in terms of multiple powers of z based on θ
d.) express cos3θ sin4 θ in terms of multiple angles.
(2.)
a.) Let z =z1/z2 where z1 = tan θ + i and z2 = z1. Find an expression for z^n with n ∈ N.
b.) Let z = cos θ - i(1 + sin θ). Determine
2z + i/-1 - iz
a.)derive the 4th roots of w =-8i
b.) express cos(4θ) and sin(5θ) in terms of powers of cos θ and sin θ
c.) expand cos^6 θ in terms of multiple powers of z based on θ
d.) express cos3θ sin4 θ in terms of multiple angles.
(2.)
a.) Let z =z1/z2 where z1 = tan θ + i and z2 = z1. Find an expression for z^n with n ∈ N.
b.) Let z = cos θ - i(1 + sin θ). Determine
2z + i/-1 - iz
Linear Algebra
Determine for which value (s) of λ the real part of z =1+λi /1- λi equals zero
Linear Algebra
Find the roots of the equation
1.)z^4 + 4 = 0 and z^4 - 4 = 0
2. ) Additional Exercises for practice are given below.
Find the roots of
(a) z^8-16 = 0
(b) z^8 + 16 = 0.
1.)z^4 + 4 = 0 and z^4 - 4 = 0
2. ) Additional Exercises for practice are given below.
Find the roots of
(a) z^8-16 = 0
(b) z^8 + 16 = 0.
Linear Algebra
1.) Determine the complex numbers i^2666 and i^145
2.) Let z1 =-i/-1+I and z2 =1+i/ 1- i. Express z1z3/z2, z1z2/z3, and z1/z3z2 in both polar and standard forms.
3.)Additional Exercises for practice:
Express z1 =-i, z2 =-1-i√3, and z3 = -√3 + i in polar form and use your results to find
(z4/3)
/z2/1 z -1/ 2 .
Find the roots of the polynomials below.
(a) P(z) = z2 + a for a > 0
(b) P(z) = z3-z2 + z-1.
(c) Find the roots of z3-1
(d) Find in standard forms, the cube roots of 8-8i
(e) Let w = 1 + i. Solve for the complex number z from the equation z^4 = w3.
4.)Find the value(s) for λ so that α = i is a root of P(z) = z^2 + λz-6.
2.) Let z1 =-i/-1+I and z2 =1+i/ 1- i. Express z1z3/z2, z1z2/z3, and z1/z3z2 in both polar and standard forms.
3.)Additional Exercises for practice:
Express z1 =-i, z2 =-1-i√3, and z3 = -√3 + i in polar form and use your results to find
(z4/3)
/z2/1 z -1/ 2 .
Find the roots of the polynomials below.
(a) P(z) = z2 + a for a > 0
(b) P(z) = z3-z2 + z-1.
(c) Find the roots of z3-1
(d) Find in standard forms, the cube roots of 8-8i
(e) Let w = 1 + i. Solve for the complex number z from the equation z^4 = w3.
4.)Find the value(s) for λ so that α = i is a root of P(z) = z^2 + λz-6.
Linear Algebra
1.
(a) Find a and b such that - 3ai-( - 1-i)b =3a-2bi.
(b) Let z1 = 12 + 5i and z2 = (3-2i)(2 + λi). Find λ without resorting to division such that z2 = z1.
2
Let z =2 + 3i and z^t = 5 - 4i. Determine the complex numbers
(a) z^2 -
zz^t
(b)1/2(z + z)^2
(c)1/2 [z-z] + [(- 1-z^t)]^2.
(a) Find a and b such that - 3ai-( - 1-i)b =3a-2bi.
(b) Let z1 = 12 + 5i and z2 = (3-2i)(2 + λi). Find λ without resorting to division such that z2 = z1.
2
Let z =2 + 3i and z^t = 5 - 4i. Determine the complex numbers
(a) z^2 -
zz^t
(b)1/2(z + z)^2
(c)1/2 [z-z] + [(- 1-z^t)]^2.
Linear Algebra
Find a and b such that -3ai-(-1 - i) b =3a - 2bi
Linear Algebra
Let z=z1/z2 where z1=tantheta +i and z2=z1. Find an expression for zn with n E N
Linear Algebra
explain how you will practice differentiations instruction when teaching about the topic of fraction by focusing on these three elements; content, process and product
Linear Algebra
Find the eigenvalues and an eigenvector per eigenvalue of the matrix
A= [ 0 1 1]
[ 3 -4 -3]
[ -5 7 6]
Is A is diagonalisable ? Give reason your answer.
A= [ 0 1 1]
[ 3 -4 -3]
[ -5 7 6]
Is A is diagonalisable ? Give reason your answer.
Linear Algebra
i. Let A= [ 5 0 0]
[ 1 5 0]
[ 0 1 5]
Find a column vector X for which A= cX, for some c belong to R.
ii. Give an example with justification, of a skew-hermitian operator on C^2
[ 1 5 0]
[ 0 1 5]
Find a column vector X for which A= cX, for some c belong to R.
ii. Give an example with justification, of a skew-hermitian operator on C^2
