Felicia Hall
HW # 3
Question 38. four 40-W lightbulbs, five 60-W bulbs, and six 75-W bulbs.
Suppose that three bulbs are randomly selected.
a. What is the probability that exactly two of the selected bulbs are rated 75-W?
( 6 )( 9 )
2 1 159
=
=0.296
P( 2-
MA 511, Session 36
Quadratic Forms
In multivariate calculus, an important problem is to
nd maxima and minima of a function of several
variables. Let us consider the case where the function F depends on 2 variables:
z = F (x, y ).
We want to understand the
MA 511, Session 31
Complex Matrices
Let us recall that the conjugate of the complex
number z = a + bi is the complex number z = a bi,
and its modulus or absolute value is the non-negative
real number |z | = a2 + b2 .
A number of the form +i can be put in
MA 511, Session 32
Similarity Transformations
Let A be a n n matrix. If S is a nonsingular n n matrix, then A S 1 AS is called a
similarity transformation.
We saw that this is related to changes of variables
in systems of dierential equations: Suppose B =
MA 511, Session 33
Jordan Forms
Let A be a n n matrix, and p() its characteristic polynomial. Let 1 , . . . , k be distinct roots of
p(), having multiplicities n1 , . . . , nk , respectively.
Let gi = dim Si be dimension of the eigenspace
Si . Then ni is
MA 511, Session 30
Stability of Solutions of Linear Systems of ODEs
Let A be a n n matrix. Then eAt is a fundamental matrix for du = Au. Any solution of this
dt
system has the form
u(t) = eAt c.
If A is diagonalizable, the components of any solution are l
MA 511, Session 29
Linear Systems of ODEs and Matrix Exponentials
Let A be a n n matrix. We want to solve the
linear, homogeneous system of ODEs with constant
coecients
dx
= Ax.
dt
Assume A is diagonalizable, S 1 AS = . We want
to show that a simple chang
MA 511, Session 27
Eigenvalues and Eigenvectors
Let A be a n n matrix.
Denition: We say the number (real or complex)
is an eigenvalue of A if the matrix A I is singular, that is, det(A I) = 0. In that case, the
system (A I)x = 0 has a non-trivial (nonzero
MA 511, Session 24
Formulas for Determinants
Let A be a n n matrix. Then we may write
P A = L U , where L, P, U Mnn are, respectively,
a lower triangular matrix with 1s on the main diagonal, a permutation matrix (with determinant equal
to 1), and an upper
MA 511, Session 28
Diagonalization
Let A be a n n matrix. Suppose that A has n
linearly independent eigenvectors, v1 , . . . , vn , corresponding to the eigenvalues 1 , . . . , n , respectively.
Let us dene the eigenvalue matrix and an
eigenvector matrix
MA 511, Session 26
Review
Example: Find a unit vector v orthogonal to the
row space of
123
A = 4 5 6
789
Solution: Since the null space is the orthogonal complement of the row space, any vector orthogonal to
the latter lies in the former. Thus, we seek a
MA 511, Session 25
Applications of Determinants
A formula for A1
Let us dene the matrix of cofactors
C
C12 . . . C1n
11
C21 C22 . . . C2n
C=
Cn1 Cn2 . . . Cnn
Theorem: A
1
1
CT
=
det A
Proof: Let us see that A C T = (det A) I. For the
diagonal entries,
MA 511, Session 34
Review
1) Let us revisit the coecients of the characteristic
polynomial. We have
p() = det(A I) = an n + an1 n1 + + a0 ,
and we already know that
an = (1)n , an1 = (1)n1 tr A, a0 = det A.
We also know that we can factor a polynomial usi
MA 511, Session 37
Quadratic Forms and Quadrics
Consider the general quadratic form in n variables
n
n
f (x1 , . . . , xn ) = i=1 j =1 aij xi xj = xT Ax,
where A = (aij ).
Example: Let
1
f (x1 , x2 , x3 ) = xT 4
7
2
5
8
3
6x
9
= x2 + 2x1 x2 + 3x1 x3 + 4x2
(I)
Math 511 Midterm
July 11, 2014 Name
1 2 4
1. Let A = 1 3 5 .
2 3 9
(1) (3 points) Factor A = LU , Where L is a lower triangular matrix and U
is an upper triangular matrix.
(2) (3 points) Find A4.
2
(3) (3 points) Solve the linear system Ax = < 1 ) .
MA 511, Session 38
The Finite Element Method
We have been studying positive denite forms
F (x) = xT Ax = (x, Ax), x Rn , A symmetric.
We have seen that in abstract vector spaces we have
linear operators and scalar products. Thus, if V is
an abstract vecto
MA 511, Session 21
The Fast Fourier Transform (FFT)
We have seen that the rst step to compute the
truncated Fourier series from a sampled digital signal
digital sampling of signal truncated Fourier series
(in complex form)
y=
y0
c =
yn1
c0
,
cn1
as well
MA 511, Session 22
More on Vector Subspaces
Let V and W be subspaces of U , and let us dene
their sum,
V + W = cfw_u = v + w U, where v V, w W .
It is easy to see that the intersection of V and
W , V W , is a subspace of U , while, in general, their
union
MA 511, Session 23
Determinants
Consider the vector space V of n n matrices
with real coecients, Mnn . We want to dene a
very useful function
det : V R
that has the following three properties:
(1) it is linear on the rst row;
(2) it is multiplied by 1 whe
MA 511, Session 6
Vector Spaces
One of the main features of the set of n-component
column vectors Rn is that there are two algebraic
operations on the set, addition of vectors and scalar
multiplication of a number by a vector:
If u, v Rn and c R (c is a s