Homeworks to review for post 12-week portion of final exam
- HW09/HW10 & Lectures 25 through 29 -
divisibility, the division algorithm (quotient and remainder), calculating with $\mathbb{Z}_n$
gcd, the euclidean gcd algorithm (calculating gcds),
multiplicative inverses in $\mathbb{Z}_n$
- HW11 & Lectures 30, 31 - vectors, vector spaces, operations on vectors
(i.e. +, -, scalar multiplication and dot product), linear
combinations of vectors
- HW12 & Lecture 32 through 37 - Matricies, matrix-vector product, elementary
row operations, Gaussian elimination, back substitution, row
echelon form, solving $A\cdot\boldsymbol{x} = \boldsymbol{0}$,
solving $A\cdot\boldsymbol{x} = \boldsymbol{b}$ (augmented matrix),
matrix products, linear transformations.
- HW13 & Lecture 40 / 41 activities -
looking at a recurrence relation and
calculating the first several elements of the sequence,
given a recurrence and a
proposed closed form, give an inductive proof that the
closed form is correct, given a first order recurrence,
expand and guess a closed form, given a recurrence relation
be able to write a simple C++ function that computes the
sequence element at a particular index.
Last few classes material
- Know what it means for a function described by a
recurrence relation to be well defined.
So if I ask why is $d(\cdot)$ described by
$$
d(n) = \frac{1}{n-3} d(n-1), \ \ d(0) = 1
$$
not well defined? You should be able to tell me it's
because we have division by zero if we try to evaluate $d(3)$.
(Class 40)
-
Know the closed forms for the sums
$1+2+3+\ldots + n$ and
$1+r+r^2+\ldots + r^n$.
(Class 39)
-
Know what "$\Sigma$" notation for sums means, and what the
"$\Pi$" notation for product means. For example,
if I give you $\Sigma_{i=3}^5 i^2$, you should be able to
tell
me that that means $3^2 + 4^2 + 5^2$.
(Class 41)
Know how to break complex sums into pieces as in HW13.
-
Be able to identify the order of a recurrence.
(Class 42)
The structure of the final exam
The final exam will heavily emphasize the post 12-week
material. However, there will be questions from the earlier
portions of the course, and all of the 6 and 12 week exam material is
fair game. So, for example, I may give you a propositional
logic formula and values for variables and ask you to evaluate
the formula. Or I may give you a first order logic proof
without justifications for the steps, and ask you to provide
those missing justifications. Or I may give you an even/odd
statement to prove. So review the 6 and 12 week exams, and
review the old HWs and in-class activities.