Lecture 3 Notes
u Properties satisfied by a line through the origin (one-dimensional case.
u A directed arrow from the origin (v) on the line, when scaled by a constant
(c) remains on the line
u Two directed arrows (u and v) on the line can be added to cr
Chapter 1-4 Cumulative Practice Answers
1. Over the period between 1970 and 2000, the percentage of the American population
in poverty each year has been about 12%. This percentage is based on income figures
from the March Current Population Survey and th
Math 121 Practice Midterm 1
No calculator allowed. Please show your work.
(1) True or False? Justify your answer.
(a) The equation Ax = b is consistent if the augmented matrix [ A b ] has a pivot posiiton
in every row.
(b) A homogeneous equation is always
Math 121 Practice Midterm 1 Solutions
No calculator allowed. Please show your work.
(1) True or False? Justify your answer.
(a) The equation Ax = b is consistent if the augmented matrix [ A b ] has a pivot positon
in every row.
Solution. False. The equati
2/13 Homework Solutions
Section 1.7
16. The second vector is obtained by rescaling the rst vector by 3 , thus the vectors are linearly dependent.
2
18. There are more vectors than entries in each vector. By Theorem 8, the vectors are linearly dependent.
1
2/6 Homework Solutions
Section 1.4
12. The augmented matrix is
1
3
5
2
4
2
1
2
3
1
2
3
Row reducing yeilds
1 2 1 1
1 2 1 1
12
0 2 1 5 0 2 1 5 0 2
0 8 8 8
0 0 4 12
00
4
So the solution is x = 4 .
3
14. The vector u is in the span of the columns of A if and
1/30 Homework Solutions
Section 1.3
11. The vector b is a linear combination of a1 , a2 , and a3 if and only if the linear system with augmented matrix
[ a1 a2 a3 b ] is consistent. We row reduce as follows:
105
2
1052
1052
2 1 6 1 0 1 4 3 0 1 4 3
028
6
0
Lecture 6 Notes
u Inner Product (aTx) <= Product of Norms (i.e. |a|x|)
u Projection length <= Product of Individual Lengths
u This is the Schwartz Inequality!
u Equality happens when a and x are in the same direction (i.e. cos = 1,
when = 0)
u Application
Lecture 4 Notes
Projects a function onto a infinite number of orthonormal basis functions: (ej or
n
ej2
), and adds the results up (to get an equivalent representation in the
frequency domain).
CDMA codes are orthogonal, and projecting the composite rece
Goldstein 1st
Gabe Goldstein
Mrs. Hammer
1st
Beta Flight Reflection
1. The most difficult part about making the parachute was cutting out the bag.
Also getting it in the best position possible. Another part was cutting string
perfect and aligned straight.