Solutions to Assignment 4
.(x+y) - (x-y).(x-y) = x.x + x.y + y.x + y.y + x.x - x.y - y.x + y.y = 2| x |2 + 2| y |2
1. | x + y | + | x - y | = (x+y)
2. x - y is orthogonal to x + y
(x-y).(x+y) = 0
x.x
Solutions to Assignment 3
1. a. This is not a subspace, since it is not closed under addition or multiplication by a scalar. For example,
are in the set but
1
0
( 0 ) and ( 1 )
( 1 ) +( 0 ) = ( 1 ) an
Assignment 1
Due:
Thursday, January 17. Nothing accepted after Tuesday, January 24. 10% off for being late, i.e.
after the 17th.
Please work by yourself on this assignment and all the other assignment
Formulas
Trigonometric Identities
sin(x+y) = sin x cos y + cos x sin y
cos(x+y) = cos x cos y - sin x sin y
sin x sin y = [ cos(x-y) - cos(x+y) ]
cos x cos y = [ cos(x-y) + cos(x+y) ]
sin x cos y = [
Solutions to Assignment 2
1 4 1
3 8 0
4 9 2
1. Start out with A(0) = A =
1 0 0
1 0 0
0 0 1
0 0 1
and L(0) = I = 0 1 0 and P(0) = I = 0 1 0 . Note that P(0)A = L(0)A(0). The largest
element in absolute
Solutions to Assignment 9
1. a. Let be an eigenvalue of R with eigenvector v. Since R2 = I on has v = Iv = R2v = RRv = Rv = Rv = v = 2v. So
v = 2v or (2 1)v = 0. Since v 0 this means 2 1 = 0 or 2 = 1
Solutions to Assignment 8
1. a. A v = AAv = Av = Av = v = v.
2
2
b. Let v be an eigenvector of A2 for the eigenvalue . Then (A2 - I)v = 0. So (A - I)(A + I)v = 0. If (A + I)v = 0, then
v is an eigenve