This preview shows pages 1–2. Sign up to view the full content.
Name:
Student ID:
Score:
/100
( +
/15 bonus)
1. Indicate whether each of the following statements is true or false. (2 points each).
(a) Every linear transformation may be represented as a matrix.
(b) An orthonormal basis can always be deﬁned for any vector space with a deﬁned
inner product.
(c) If all entries of a real, symmetric,
n
square matrix,
A
, are positive, then
~x
T
A
~x >
0
for any
n
×
1 vector
~x
6
= 0.
(d) The following matrix is invertible:
0 3 0 2
7 1 0 9
6 9 2 4
0 7 0 0
(e) The eigenvalue decomposition of a matrix,
A
(given by
A
=
PDP

1
where
D
is
diagonal) is unique.
(f) An
n
square matrix always has
n
real eigenvalues if and only if the matrix is
invertible.
(g) All functions that are continuous everywhere in [a,b] are diﬀerentiable everywhere
in (a,b).
(h) A inﬁnite series,
∑
∞
i
=1
u
i
, converges if and only if the corresponding sequence
{
u
i
}
converges to zero (
i.e.
lim
i
→∞
u
i
= 0).
(i) lim
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 02/09/2010 for the course AMS 510 taught by Professor Feinberg,e during the Fall '08 term at SUNY Stony Brook.
 Fall '08
 Feinberg,E
 Applied Mathematics

Click to edit the document details