Linear Algebra Answers

For the subspace W= the plane 3x-y+4z-0, of R

Suppose U and W are subspaces of a vector space V, where dimR(V)=8. Suppose dimR(U)= 4, and dimR(W)= 5. What are the possible values of dimR(U∩ W)?

a. Find the orthogonal and normal canonical forms of 2y^2-2yz+2zx-2xy.
b. The operation,* defined by a*b= sin(ab), is a binary operation on N
True or false with full explanation

a. Can [ i ] be a column of a unitary matrix?
[-i ]
Justify your answer
b. Check whether or not the following linear system is consistent:
3+t = x+y+z, 2+2t= x+2y-z, 4-t= x-y +4z

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

Determine for which value (s) of λ the real part of z =1+λi /1- λi equals zero

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.