Download 21-241 (Fall 15) Problems for Review Session (Sep 27, 2015) 1.

21-241 (Fall 15)
Problems for Review Session (Sep 27, 2015)
Page 1 of 3
1. (a) Fill in the blanks.
A B
¬A
A∧B
A∨B
A =⇒ B
A ⇐⇒ B
T T
T F
F T
F F
(b) State the definition of even and odd used in class.
(c) In each of the following statements, circle the terms (e.g. x, y) that the bolded quantifiers
refer to.
i. ∀x ∈ Z, ∃y ∈ Z, P (x, y) =⇒
Q(x)
ii. ∀x ∈ Z,
∃y ∈ Z, P (x, y) ∨ ∃y ∈ Z, Q(x, y)
(d) Negate the following statements:
i. ∀x, P (x)
ii. ∃x, P (x)
iii. P ∧ R
iv. P ∨ R
v. P =⇒ R
vi. P ⇐⇒ R
(e) Consider the statement:
∀x ∈ Z, ∃y ∈ Z, x = 2y + 1 =⇒ ∃y ∈ Z, 5x2 = 4y + 1
If we denote the statement as ∀x ∈ Z, P (x), write down what P (x) is.
If we denote P (x) as Q(x) =⇒ R(x), write down what Q(x) and R(x) are.
What is the negation of R(x)?
Using the above parts (or not), negate the original statement. Be sure to show all
your working.
(f) Prove or disprove the statement in 1(e).
i.
ii.
iii.
iv.
2. In this problem, we are interested in properties of matrix transposition and symmetric matrices. A matrix is said to be symmetric if and only if A = At .
4 3 (a) Let C =
be a 2 × 2 matrix. If D = cji + j ij , write out the matrix D.
2 1
(b) Using the [ ]ij notation, prove that (At )t = A.
(c) Using the [ ]ij notation, prove that (AB)t = B t At . (Hint: Let A = [aij ]ij . Then what
is At ? )
(d) Using 2(a), show that AAt and At A are symmetric matrices.
(e) Prove or disprove: AAt = At A.
(f) State the most general condition(s) under which A + At is symmetric. Then prove that
A + At is symmetric with these/this condition(s).
3. (a) Solve the following linear systems by Gaussian Elimination.
a + 2b + 3c = 1
d + 2e + 3f = 0
g + 2h + 3j = 0
b + 4c = 0
e + 4f = 1
h + 4j = 0
5a + 6b = 0
5d + 6e = 0
5g + 6h = 1
(Hint: Solve them all together.)
21-241 (Fall 15)
Problems for Review Session (Sep 27, 2015)
(b) Compute
(c) Compute
1 2 3 !
0 1 4
5 6 0
a d g !
a d g !
b e h .
c f j
1 2 3 !
b e h
c f j
0 1 4
5 6 0
Page 2 of 3
.
2 2 4 2 !
3 x −1 y
has
−1 0 5 1
(d) Find values of x and y for which
i. a unique solution
ii. infinitely many solutions
iii. no solution.
4. (a) State all the axioms concerning vector spaces.
(b) Let M be the set of all 2 × 2 square matrices with real entries. Show that M is a vector
space under matrix addition and scalar multiplication over the reals.
(c) State the definition of a linear transformation as in the lecture notes/textbook.
(d) Prove that the following is an equivalent definition for linear transformation:
Let T : U → V be a function where U and V are vector spaces. T is a linear transformation if for x, y ∈ U and scalar c, T (x + cy) = T (x) + cT (y).
" 13 #
" 1 # " 2 #
" 4 #
9
as a linear combination of −1 , 0 , and 3 .
(e) Express
22
1
1
5
(f) Let T : R3 → M be a linear transformation such that
" 1 #
" 2 #
" 4 #
1 1 2 0 4 −3 , T( 0 ) =
, T( 3 ) =
.
T ( −1 ) =
1 2
1 3
5 9
1
1
5
" 13 #
Find T ( 9 ).
22
" x #
x −y (g) You may have observed that T :
→ M is actually defined by y 7→
.
z x+z
z
Prove that this is indeed a linear transformation.
R3
5. (a) Fill in the blanks.
Object
Normal form
line in R2
line in R3
plane in R3
General form
Vector form
(b) Write down a system of 2 linear equations that gives the plane:
P1 :
" 2 #
" 1 #
" 3 #
x= 4 +s 1
+ t 4 , s, t ∈ R
1
−1
−2
Parametric form
21-241 (Fall 15)
Problems for Review Session (Sep 27, 2015)
Page 3 of 3
" 3 #
(c) Find the equation of the line ` that passes through 2 and is perpendicular to the
5
" 2 #
normal of the plane and the direction 4 . Express your final answer in the parametric
1
form.
6. True or False? If false, rectify the statement.
(a) Differentiation is a linear transformation.
(b) To prove that R2 restricted to x ≥ y is not a vector space, we want to show that the
closure property does not hold. It suffices to show that there is a v = (xv , yv ) with
xv ≥ yv , and its inverse −v = (−xv , −yv ) is not in the space since −xv ≤ −yv .
(c) Given A and B are matrices, and AB is a well-defined matrix multiplication, we can
conclude that BA is also a well-defined matrix multiplication.
(d) The dot product operation on two vectors in Rn can be expressed as the matrix multiplication of a 1 × n matrix and an n × 1 matrix.
(e) If A and B are square n × n matrices, then A + B = B + A and AB = BA.
Bonus. We have never actually proven that −u = (−1)u. Prove it.
(Hint: If that statement seems very obvious, here is the question rephrased: ‘why is the inverse
of a vector v equals to −1 times v?’ )
END OF REVIEW PAPER