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Exam Review 1 Spring 16, 21-241: Matrices and Linear Transformations

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Exam Review 1 Spring 16, 21-241: Matrices and Linear Transformations
Exam Review 1
Spring 16, 21-241: Matrices and Linear Transformations
February 12, 2016
Abstract
[TOPICS COVERED] The fundamental problem in linear algebra. Solving systems of linear equations. Matrices and Gaussian elimination. Logic. Proof techniques. Vector spaces.
Geometry of linear equations. Linear transformations. Matrix multiplication. Matrix algebra.
1
Some problems
1. (a) Fill
A
T
T
F
F
in the blanks.
B
¬A
T
F
T
F
A∧B
A∨B
A =⇒ B
A ⇐⇒ B
(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.
ii.
iii.
iv.
v.
vi.
∀x, P (x)
∃x, P (x)
P ∧R
P ∨R
P =⇒ R
P ⇐⇒ R
(e) Consider the statement:
∀x ∈ Z, ∃y ∈ Z, x = 2y + 1 =⇒ ∃y ∈ Z, 5x2 = 4y + 1
i.
ii.
iii.
iv.
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.
1
(f) Prove or disprove the statement in 1(e).
2. The following augmented matrices are derived from solving a system of linear equations in
R4 , with the columns corresponding to x1 , x2 , x3 , x4 respectively. Determine the set of all
solutions for each system.


1 0 0 0 0
 0 1 0 0 2 

(a) 
 0 0 1 0 4 
0 0 0 1 8


0 1 0 1 4
(b)  0 0 1 2 4 
0 0 0 0 0
1 0 0 0 3
(c)
0 0 1 0 5
0 1 0 2 3
(d)
0 0 0 0 1


1 0 2 0 2
(e)  0 1 3 0 3 
0 0 0 1 5
0 0 0 0 0
(f)
0 0 0 0 0
(g) 0 1 2 3 4
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.)
1 2 3 ! a d g !
0 1 4
b e h .
(b) Compute
5 6 0
c f j
a d g ! 1 2 3 !
(c) Compute
b e h
c f j
0 1 4
5 6 0
.
4. Consider the following system of linear equations:
3x + 2y = 9
2x − y = −1
(a) Solve the system.
(b) Express your solution in a row picture.
(c) Express your solution in a column picture.
Page 2
5. Let M be the set of all 2 × 2 square matrices. Define the following operations:
⊕ : M × M → M defined by A ⊕ B = A + B
and
: R × M → M defined by c A = cA
(a) We claim that {M, R, ⊕, } (i.e. the vector space with vectors in M and scalars in R
with addition ⊕ and scalar multiplication ) is a vector space. In order to prove this,
we need to verify all axioms, which we shall not do all of that. Instead, verify only that
the following axioms:
i. (c + d)v = cv + dv
ii. (cd)v = c(dv)
" 13 #
(b) Express
9
22
" 4 #
" 1 # " 2 #
as a linear combination of −1 , 0 , and 3 .
5
1
1
(c) Let T : R3 → M be a linear transformation such that
" 1 #
" 2 #
" 4 #
1 1 2 0 4 −3 T ( −1 ) =
, T( 0 ) =
, T( 3 ) =
.
1 2
1 3
5 9
1
1
5
" 13 #
Find T ( 9 ).
22
" x #
x −y .
(d) 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
6. Consider the polynomial space P2 . Let p(x) = a + bx + cx2 ∈ P2 .
(a) What is the zero vector?
(b) What is the inverse of p(x)?
(c) Is p(x + 1) in P2 ?
(d) Is p(5) in P2 ?
(e) When is p(x2 ) in P2 ?
7. In this problem, x, y, z, 0 (zero vector) are n × 1 vectors, Q is an n × n matrix, c is a scalar.
Circle the following expressions that should never appear on any of your workings.
(a) x · y = xT y
(b) (x · y)z = (x · z)y
(c) cx = x[ c ]
(d) (x · y)z = (xT y)z
(e) (x · y)z = (zxT )y
(f)
y·x
z·x
=
y
z
Page 3
(g) 0 = 0
8. Compute the following product of matrices:

 
1
1
 2 
 3
  0 2 4 6 
 5
 3 
7
4




9. Let {Ak }k∈N be a sequence of m × n matrices. Using induction, prove that for all ` ∈ N,
`
`
X
X
−(
Ai ) =
(−Ai ).
i=1
i=1
10. Let A and B be n × n matrices. Express each column of AB as a linear combination of the
columns of A.
2
Some harder problems
11. Find values of x and y for which
2 2 4 2 !
3 x −1 y
has
−1 0 5 1
(a) a unique solution
(b) infinitely many solutions
(c) no solution.
12. Consider the following plane in R3 :
P1 :
" 2 #
" 1 #
" 3 #
+ t 4 , s, t ∈ R
x= 4 +s 1
1
−1
−2
(a) Is P1 expressed in general form, normal form, vector form, or parametric form?
(b) Express P1 in general form.
" 3 #
(c) Find the equation of the line ` that passes through 2 and is perpendicular to the
5
" 2 #
4 . Express your final answer in the
normal of the plane P1 and the direction
1
parametric form.
13. Prove that in a vector space, the following properties hold:
(a) 0v = 0.
(b) (−1)v = −v.
(c) The zero vector is unique.
Page 4
(d) The additive inverse is unique.
14. Let P be the powerset of N, the set of natural numbers. Consider the vector space {P, F, 4, }
where F consists of {0, 1} with addition defined modulo 2, 4 is the symmetric set difference,
and is such that 1 S = S and 0 S is the zero vector.
(a) Let S ∈ P . What must 1 S be?
(b) What is the zero vector?
(c) Let S ∈ P . What is −S?
(d) Prove that the following axiom
c(u + v) = cu + cv
is satisfied.
15. Let V and W be vector spaces. We call a function f : V → W a LT if f satisfies the condition
f (cu + v) = cf (u) + f (v) for scalar c and vectors u and v. Prove that f is LT if and only if
it is a linear transformation.
16. Consider the function
T : F → F defined by T (f (x)) = f (x2 )
Is T a linear transformation? Prove or disprove.
17. 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 [
is At ? )
]ij notation, prove that (AB)t = B t At . (Hint: Let A = [aij ]ij . Then what
(d) Using what you have done above, 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).
18. A matrix [ai,j ]ij is called a diagonal matrix if it is of the form
λi
if i = j
ai,j =
0 otherwise
Suppose A is a diagonal square matrix. Prove by induction that Ak = [akij ]ij for k ∈ N.
Page 5
3
Just for fun
19. A museum has a main gallery and a north wing, south wing, east wing, and west wing. Each
wing is only connected to the main gallery (i.e. the museum is a cross shape). The lighting in
each room takes on only two states: ON or OFF. There is one light switch per room. Because
the engineers are not from CMU, they completely messed up the wiring such that flipping a
switch in one room switches the states of the lights in the room and the adjacent rooms.
An example of this is in the following diagram:



OFF
 ON ON OF F  −→  ON
ON
ON
OF F
ON

OF F 
when the switch in the north wing is flipped.
The president is on his way to visit the museum and the curator want to turn on all the lights
to make the museum look presentable. Alas, he doesn’t know how. However, he spots you
admiring a great piece of art. Knowing you are from CMU he places his faith in you and
enlists your help.
Whatever the state of the lights in the various rooms, can you always turn them all on? Prove
or disprove.
GOOD LUCK ON THE EXAM!
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