8.1
Similarity and triangularizability
Remark 8.6. If L : U U is linear and we are given bases U , V for U, then
V
L
V
U
=I
L
U
I
V
U
U
V
.
Definition 8.7. For A, B M (F), we say A and B are similar when B = P 1AP for some invertible matrix P M (F).
nn
No
MATH 245 Linear Algebra 2, Assignment 5
Due Mon July 25
3 2 4
1: (a) Let A = 2 0 2 . Find an orthogonal matrix P and a diagonal matrix D such that P tAP = D.
4 2 3
(b) Let A =
2+i
i
1+i
. Find a unitary matrix P and an upper-triangular matrix T so that P
MATH 245 Linear Algebra 2, Assignment 4
Due Mon July 18
1: For 0 = u R3 and R, let Ru, : R3 R3 denote the rotation about the vector u by the angle (where
the direction of rotation is determined by the right-hand rule: the right thumb points in the directi
MATH 245 Linear Algebra 2, Assignment 3
Due Mon July 4
1
2
1
1
0
1
3
1
1: Let u1 = , u2 = , u3 = and x = . Let U = cfw_u1 , u2 , u3 and let U = Span U . Find
1
1
2
7
1
0
1
3
Proj (x) in the following three ways.
U
(a) Let A = u1 , u2 , u3 M43 then use t
MATH 245 Linear Algebra 2, Assignment 2
Due Mon June 20
1
1: Let p1 (x) = x 1, p2 (x) = 1 (x2 3x) and p3 (x) = 2 (x3 3x2 + 2). Find the polynomial f Spancfw_p1 , p2 , p3
2
5
f (ai ) bi
which minimizes the sum
2
for the 5 points (ai , bi ) given below
i=1
MATH 245 Linear Algebra 2, Assignment 1
Due Fri May 27
1: (a) Let A be a set. For each A, let p Rn , let U be a vector space in Rn , and let P =
p + U .
A
Show that if P is not empty then it is an ane space in Rn .
231
1
1 2 0
3
1 2 1 1 2
1
(b) Let p = 1
MATH 245 Linear Algebra 2, Solutions to Assignment 6
1: (a) For the quadratic curve 7x2 + 8xy + y 2 + 5 = 0, nd the coordinates of each vertex, nd the equation of
each asymptote, and sketch the curve.
x
y
Solution: Let K (x, y ) = 7x2 + 8xy + y 2 . Note t
MATH 245 Linear Algebra 2, Solutions to Assignment 5
3 2 4
1: (a) Let A = 2 0 2 . Find an orthogonal matrix P and a diagonal matrix D such that P tAP = D.
4 2 3
Solution: The characteristic polynomial of A is
fA (t) =
3t 2
2 t
4 2
4
2
3t
= t(t3 6t + 9) +
MATH 245 Linear Algebra 2, Solutions to Assignment 4
1: For 0 = u R3 and R, let Ru, : R3 R3 denote the rotation about the vector u by the angle (where
the direction of rotation is determined by the right-hand rule: the right thumb points in the direction
MATH 245 Linear Algebra 2, Assignment 6
Not to be handed in
1: (a) For the quadratic curve 7x2 + 8xy + y 2 + 5 = 0, nd the coordinates of each vertex, nd the equation of
each asymptote, and sketch the curve.
(b) For the real quadratic form K (x, y, z ) =
7.2
Dual spaces and quotient spaces
Definition 7.20. Let W b e a vector space over a field F and let U b e a subspace of W. Then we define the quotient space
W/U to be the vector space
W /U = cfw_x + U : x
W
with addition given by
and the zero given by 0
Definition 1.1. An affine space in Rn is a set of the form p + U = cfw_p + u | u U for some p oint p Rn and some
vector space U in Rn .
Theorem 1.2. Let P = p + U and Q = q + U b e two affine spaces. We have p + U q + V if and only if q p V and U
V.
Proo
Alternate pro of : Suppose L = L. Let b e any eigenvalue of L and let u b e an eigenvector for . Then
hu, ui = hu, ui = hL(u), ui = hu, L ui = hu, L(u)i = hu, ui =
hu, ui
therefore = since hu, ui = 0, therefore R.
Since the eigenvalues are all real, fL s
9
Bilinear and quadratic forms
9.1
Bilinear forms
Definition 9.1. Let U, V, W b e vector spaces over any field F. Then a map F : U V W is called bilinear when the following
are satisfied:
F(u + v , w) = F(u, w) + F(v, w )
F(u, v + w) = F(u, v ) + F(u, w
Theorem 9.12 (Sylvesters Law of Inertia). Let U b e an n-dimensional vector space over R. Let F : U U R be a symmetric
bilinear form on U. Let U and V b e bases for U such that
I
I
Ik
I`
r k
r `
U=
V=
0
F
and
F
nr
0
n r
then k = `. The number k is called
9.2
Quadratic forms
Definition 9.17. Let U b e a vector space over a field F. A quadratic form if a map K : U F given by
K(u) = F(u, u)
for some symmetric bilinear form F on U.
Remark 9.18. When U is finite dimensional and U is a basis for U we write
U
K
Theorem 2.14. Let A M
(R). Then null(A A) = null(A).
kn
t
t
Proof. For x R , x null(A) implies Ax = 0, so A Ax = 0. Therefore x null(A A).
t
t
tt
t
2
If x null(A A) then A Ax = 0, then x A Ax = 0, so (Ax) (Ax) = 0, so (Ax) (Ax) = 0, so |Ax|
= 0, so |Ax| =
Proof. Easy.
Definition 2.3. For u R
n
we define the length (or norm) of u to be
|u| := u
u.
Theorem 2.4 (Properties of Length). Length satisfies, for all u, v, w R
n
and t R:
1. [positive definite] |u| 0, holding with equality if and only if u = 0.
2. [
Figure 3: The medial hyp erplane M
1,2
(shaded) in a 3-simplex.
n
Definition 1.17. Let [a , . . . , a ] be an `-simplex in R . For 0 j < k `, the (j, k) medial
0
`
hyp erplane M
is defined by
j,k
M
i.e. the affine span of the points a
i
1
= ha , (a + a
j,
1
Affine spaces
A vector space in R
spancfw_v , . . . , v .
1
`
n
is a set of the form U =
n
Definition n .1. An affine space in R
1
is n set of the form p + U = cfw_p + u | u U for some
a
point p R and some vector space U in R .
Theorem 1.2. Let P = p +
MATH 245 Linear Algebra 2, Solutions to Assignment 3
1
2
1
1
0
1
3
1
1: Let u1 = , u2 = , u3 = and x = . Let U = cfw_u1 , u2 , u3 and let U = Span U . Find
1
1
2
7
1
0
1
3
Proj (x) in the following three ways.
U
(a) Let A = u1 , u2 , u3 M43 then use the
MATH 245 Linear Algebra 2, Solutions to Assignment 2
1
1: Let p1 (x) = x 1, p2 (x) = 1 (x2 3x) and p3 (x) = 2 (x3 3x2 + 2). Find the polynomial f Spancfw_p1 , p2 , p3
2
5
f (ai ) bi
which minimizes the sum
2
for the 5 points (ai , bi ) given below
i=1
i
MATH 245 Linear Algebra 2, Solutions to Assignment 1
1: (a) Let A be a set. For each A, let p Rn , let U be a vector space in Rn , and let P =
Show that if P is not empty then it is an ane space in Rn .
p + U .
A
U . We claim that U is a vector space in R
MATH 245 Linear Algebra 2, Solutions to Assignment 2
1: Let U and V be vector spaces in Rn .
(a) Show that (U + V ) = U V .
.
Solution: Let x (U + V ) . This means that x y = 0 for all y U + V . For all u U , we also have
u U + V so that x u = 0, and so x
MATH 245 Linear Algebra 2, Solutions to Assignment 1
2
1
3
1
2
4
1: Let p =
, u1 =
, u2 = , q =
2
1
1
1
3
5
intersection of the plane x = p + t1 u1 + t2 u2 and
1
1
1
2
1
2
, v1 = and v2 =
. Find the point of
3
4
1
0
2
1
the plane x = q + s1 v1 + s2 v
MATH 245 Linear Algebra 2, Assignment 7
Not to be handed in
1
2
1: Find a singular value decomposition A = QP for the matrix A =
3
1
2
0
.
1
1
2: Let A Mnn (C) be Hermitian and positive-denite. Show that if A = QP is a singular value decomposition of A,
MATH 245 Linear Algebra 2, Assignment 6
Due Fri Dec 3
1: (a) For the quadratic curve 7x2 + 8xy + y 2 + 5 = 0, nd the coordinates of each vertex, nd the equation of
each asymptote, and sketch the curve.
(b) For the real quadratic form K (x, y, z ) = 3x2 +
MATH 245 Linear Algebra 2, Assignment 5
1
1: (a) Let A = 2
1
(b) Let A =
2
4
2
Due Wed Nov 24
1
2 . Find an orthogonal matrix P and a diagonal matrix D such that P AP = D.
1
2i 2 + i
. Find a unitary matrix P and an upper-triangular matrix T so that P AP
MATH 245 Linear Algebra 2, Assignment 4
Due Wed Nov 10
12
34
3 1
, A2 =
and A3 =
. Apply the Gram-Schmidt Procedure to the
1 0
1 2
2 4
basis U = A1 , A2 , A3 to obtain an orthonormal basis for U = Span A1 , A2 , A3 M22 .
1: (a) Let A1 =
(b) Find an orthon
MATH 245 Linear Algebra 2, Assignment 3
Due Fri Oct 15
1: (a) Find the least-squares best t quadratic f P2 for the following data points.
x i 1
yi
0
0123
2 3 2 2
(b) Let cfw_p1 , p2 , , pl be a linearly independent set of polynomials in Pm and let (x1 ,