Solutions
Math 21b, Spring 09
1. The verication that cos(nx), sin(nx), 1/ 2 form an orthonormal family is a straightforward computation, when using the identities provided. For example, cos(nx), sin(mx) =
1
1
sin(nx) sin(mx) dx = 2 cos(n m)x) cos(n + m)x
10/7/2010 FIRST HOURLY PRACTICE IV
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
n
FINAL PRACTICE V, December 17, 2010
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
10/7/2010 FIRST HOURLY PRACTICE I
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
ne
11/4/2010 SECOND HOURLY PRACTICE IV
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
10/7/2010 FIRST HOURLY PRACTICE II
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
n
10/7/2010 FIRST HOURLY PRACTICE V
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
ne
10/7/2010 FIRST HOURLY PRACTICE II
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
n
11/4/2010 SECOND HOURLY PRACTICE III
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
11/4/2010 SECOND HOURLY PRACTICE I
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
n
11/4/2010 SECOND HOURLY PRACTICE VI
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
FINAL PRACTICE I, December 17, 2010
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
10/7/2010 FIRST HOURLY PRACTICE I
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
ne
10/7/2010 FIRST HOURLY PRACTICE III
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
10/7/2010 FIRST HOURLY
Math 21b, Fall 2010
Name:
Problem 1) TF questions (20 points) No justications are needed.
1)
Start by writing your name in the above box and
check your section in the box to the left.
T
F
The rank of A1 is always equal to the rank
FINAL PRACTICE V, December 17, 2010
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
10/7/2010 FIRST HOURLY PRACTICE V
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
ne
11/4/2010 SECOND HOURLY PRACTICE II
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
FINAL PRACTICE IV, December 17, 2010
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the
[:1 ]_. so that T is indeed linear, with matrix
2
2.1.6 Note that 31 +352 [
it :1.
is 6J
2.1.14 3. By Exercise 13a, [2 3
r k] is invertible if [and only if) 211' 15 # U. or k # 2.5.
U
1
F . 2 3 _ 1 I; 3
h E} Exercise 13b. [5 k] 23:45 [_5 2].
3
FINAL PRACTICE I, December 17, 2010
Math 21b, Fall 10
Name:
Start by writing your name in the above box and
check your section in the box to the left.
Try to answer each question on the same page as
the question is asked. If needed, use the back or the