= 1. On the other hand, the intersection of V and
W is a subspace of both V and W, so its dimension is at most
the min of the dim(V) and dim(W), i.e. d = <2. Thus
d = 1 or 2 are the only possibilities.
These are realized, by the examples
V = span{e1, e2}, W = span{e2,e3,e4}
and
V = span{e1, e2}, W = span{e1,e2,e3}
respectively.
Now try the same kind of reasoning for the problem at hand.
56 Again, try a case where n is small to get intuition:
for n=2, the example
| 0 1 |
A = | |
| 0 0 |
works, with v1 = e1, Av1 = e2.
If you now just follow the hint in the book, and apply
A^[M-1} to the relation sugested, you'll get the equation
A^{m-1}(c0 v) = c0 A^{m-1}(v) = 0
which can only happen if c0 = 0, since we assumed
A^{m-1}(v) is NOT zero (this defines "nilpotent
of order m" - the first example has order m = 2).
What power of A would you apply to the relation to
get c1 = 0?
- Good luck!]
04/03 3.4 # 2, 8, 20, 32!, 36!, 40, 48, 50!, 56, 60
For the rest of the semester we plan to cover the following topics:
Linear spaces (chapter 4)
Orthogonality: Gram-Schmidt (5.1) and inner product spaces (5.5)
Determinants (6.1, 6.2)
Eigenvalues, eigenvectors (chapter 7)
04/10 4.1 #4, 8, 14!, 24, 26, 36, 40!, 50, 52, 58! [... but do it early!]
04/17 4.2 #2, 4, 6, 17!, 20, 22, 23!, 64, 65!, 78!
4.3 #2, 4, 5!, 15!, 18, 23!, 29!, 48, 49!
For those of you not familiar with differential equations,
here's where you can learn a little more:
www.math.umass.edu/~kusner/class/diffeqnotes.pdf
("Quick and Dirty Differential Equations" notes by Peter Norman,
with just a little help from yours truly ;-)
Read Chapter 5. Think about the relation between orthogonality
and linear independence. How do you find "coordinates" of a vector
with respect to an orthonormal basis? We'll discuss the Gram-
Schmidt process in detail very soon....
04/24 [Rob will be away in Philadelphia part of this week - a "guest
lecture" or review may be given by ??? - and the second MINI-MIDTERM will
happen in class either Tuesday 21 or Wednesday 22 (Monday is a holiday).]
5.1 # 6, 10, 12, 18, 28, 36!,40!, 42!
In class we will have discussed an inner product on the
polynomials in x:

is the integral of p(x)q(x) dx from -1
to 1. A basis for the quadratic polynomials is {1, x, x^2}.
Use the Gram-Schmidt process to convert this into an orthonormal
basis with respect to this inner product.
[Section 5.5 touches on this also.]
5.2 # 4, 6, 20!, 32, 34!, 40!
5.3 # 4, 10, 16!, 18!, 20!, 28, 31!, 34!, 37!
5.5 # 2, 4, 8, 9!
[Read chapters 6 and 7]
05/01 6.1 # 2, 8, 14, 24, 26, 32, 38!, 46!, 52!
6.2 # 2, 8, 12, 14, 16, 18!, 20!, 26!, 30
05/08 7.2 # 6, 12, 16, 20, 28
7.3 # 4, 16, 26
05/15 7.4 # 2, 12, 14, 40!, 48!, 58!
********************REVIEW SESSION************************
We'll plan to meet sometime to review before the final exam.
Practice Exam: http://www.gang.umass.edu/~kusner/class/pracexam2.pdf
Also, please think about this problem:
Suppose we want to compute the exponential of a matrix
exp(A) = I + A + A^2/2 + ... + A^k/k! + ...
- how does having a basis of eigenvectors for A, along with
their corresponing eigenvalues, help us?!
Practice final solutions (not sure I like the sound of that):
http://www.math.umass.edu/~kusner/prac235finalsols.pdf
HAVE A GREAT SUMMER!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!