Download Homework 5

Homework 5
Math 19620 (section 52), Winter 2016
This homework is due in class on Tuesday, February 9th. You may cite results from class as appropriate. Unless otherwise stated, you must provide a complete explanation for your solutions, not simply an
answer. You are encouraged to work together on these problems, but you must write up your solutions
independently.
You are encouraged to think about the problems marked with a (*), but they are not to be handed in.
0. (*) Read sections 3.4, 4.1-4.3.
ε. Approximately how many hours did you spend on this problem set?
1. Which of the subsets of P2 are subspaces of P2 ?
(b) (4.1.3) {p(t)|p0 (1) = p(2)}
(a) (4.1.1) {p(t)|p(0) = 2}
2. Which of the subsets of R3×3 are subspaces of R3×3 ?
(a) (4.1.6) The invertible 3 × 3 matrices.
 
1
 
(b) (4.1.10) The 3 × 3 matrices A such that 2 is in the kernel of A.
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3. (4.1.14) nLet V be the
all sequences of real numbers (see Example 4.1.5). Is the subset
space of o
W = (x0 , x1 , . . .) lim xn = 0 (of sequences that converge to 0) a subspace of V ?
n→∞
4. Find a basis for each of the linear spaces V below, and determine its dimension.
(a) (4.1.19) The (real) linear space C2 .
#
"
a b
in R2×2 with a + d = 0.
(b) (4.1.20) The space of all matrices
c d
"
#
"
1 2
0 0
(c) (4.1.30) The space of all 2 × 2 matrices A such that
A=
3 6
0 0
#
5. A function f from R to R is called even if f (−t) = f (t) for all t in R, and is called odd if f (−t) = −f (t)
for all t in R.
(a) (4.1.47) (*) Is the set even functions a subspace of F (R, R)? What about the set of odd
functions?
(b) (4.1.48) Find a basis for each of the linear spaces {f ∈ P4 |f is even} and {f ∈ P4 |f is odd}.
What are their dimensions?
6. Which of the transformations below are linear? For those that are linear, determine whether they are
isomorphisms.
(a) (4.2.2) T (M ) = 7M from R2×2 to R2×2 .
(b) (4.2.4) T (M ) = det(M ) from R2×2 to R.
"
#
2
3
(c) (4.2.10) T (M ) = P M P −1 , where P =
, from R2×2 to R2×2 .
5 7
(d) (4.2.20) T (x + iy) = x − iy from C to C.
(e) (4.2.28) T (f (t)) = f (2t) − f (t) from P2 to P2 .
(f) (4.2.30) T (f (t)) = t(f 0 (t)) from P2 to P2 .
1
7. Find the image, rank, kernel and nullity of the following linear transformations:
(a) (4.2.53) T (f (t)) = f 00 (t) + 4f 0 (t) from P2 to P2 .
R3
f (t) dt from P2 to R.
(b) (4.2.54) T (f (t)) = −2
8. (4.2.66) Find the kernel and nullity of the linear transformation T (f ) = f − f 0 from C ∞ to C ∞ . [Hint:
Recall from calculus, for which functions f (t) is f 0 (t) = f (t)?]
"
# "
# "
# "
#
1 1
1 2
2 3
1 4
9. (4.3.2) Are the matrices
,
,
,
linearly independent?
1 1
3 4
5 7
6 8
10. Find the matrix of the given linear transformation with respect to the given basis. In each case,
determine whether T is an isomorphism. If it is not, find bases for the kernel and image, and thus
determine the rank of"T . #
"
# "
# "
# "
#!
1 1
1
0
0
1
0
0
0
0
(a) (4.3.13) T (M ) =
M from R2×2 to R2×2 with basis A =
,
,
,
2 2
0 0
0 0
1 0
0 1
"
#
1 1
(b) (4.3.14) T (M ) =
M from R2×2 to R2×2 with basis B =
2 2
"
# "
# "
# "
1 0
0 1
1 0
0 1
,
,
,
−1 0
0 −1
2 0
0 2
(c) (4.3.18) T (z) = (2 + 3i)z from C to C with basis A = (1, i).
(d) (4.3.22) T (f (t)) = f 00 + 4f 0 from P2 to P2 with basis A = (1, t, t2 ).
"
#
"
#
2 0
2 0
(e) (4.3.34) T (M ) =
M −M
from R2×2 to R2×2 with basis
0 5
0 5
"
A=
# "
# "
# "
0 1
0 0
0 0
1 0
,
,
,
0 0
1 0
0 1
0 0
#!
11. (4.3.44)
(a) Find the change of basis matrix S from the basis B considered in exercise 10(b), to the standard
basis A of R2×2 considered in exercise 10(a).
(b) Verify the formula SB = AS for the matrices B and A you found in exercises 10(b) and 10(a),
respectively.
12. Let V be the plane with equation x1 + 2x2 + 3x3 in R3 . Find the
matrix,
  B,of
 each of the following
1
5
   
transformations T : V → V , with respect to the basis B =  1  , −4. Notice that V is 2
−1
1
dimensional, so B will be a 2 × 2 matrix.
 


−2 −3 1


0 −2 ~x.
(b) (4.3.57) T (~x) =  1
0
1
1
1
 
(a) (4.3.56) T (~x) = 2 × ~x.
3
" #
" #
" #
"
#
1
3
2
−2
13. (4.3.69) If A is a 2 × 2 matrix such that A
=
and A
=
show that A is similar to
2
6
1
−1
a diagonal matrix D. That is, find an invertible S such that S −1 AS = D. [Hint: Think of S as a
change of basis matrix. If a linear transformation T is diagonal with respect to a basis B = (~v1 , ~v2 ),
what can you say about T (~v1 ) and T (~v2 )?]
2
#!
14. (4.2.78) In the definition of a vector space, we usually think of the operations addition and scalar
multiplication mentioned as being the ‘usual’ operations. However, this does not need to be the
case. As long as we can pick operations satisfying the list of properties on page 167, our set will be
a vector space.
Let R+ be the set of positive real numbers. Define ‘addition’ (⊕) and ‘scalar multiplication’ () by
x ⊕ y = xy
and
k x = xk ,
where the operations on the right are the ‘usual’ operations in R. For instance, in R+ we would say
1
2 ⊕ 3 = (2)(3) = 6 and (−2) 5 = 5−2 = .
25
(a) (*) Check that R+ is a vector space with these operations. Find a basis for R+ .
(b) What is the neutral element of R+ (i.e. the element n with n ⊕ x = x for all x)? For x in R+ ,
what is the inverse of x (i.e. the element y with x ⊕ y = n)?
(c) Show that T (x) = ln(x) is a linear transformation from R+ to R (where R has the usual
operations). Is T an isomorphism?
15. (4.1.60) (*) Consider the Fibonacci sequence: (f0 , f1 , f2 , . . .) defined recursively by f0 = 0, f1 = 1
and fn = fn−2 + fn−1 for all n = 2, 3, 4, . . . [i.e. the sequence 0, 1, 1, 2, 3, 5, 8, 13, . . .]. The purpose of
this problem is to use linear algebra to find a ‘closed form’ formula for fn - i.e. an explicit formula
in terms of n, not involving recursion.
(a) Find the terms f0 , f1 , f2 , . . . , f10 .
(b) Let V be the set of all infinite sequences (from Example 4.1.5). Consider the subset W consisting of all sequences (x0 , x1 , x2 , . . .) that satisfy the recursive equation xn = xn−2 + xn−1
for all n = 2, 3, 4, . . . (so in particular, (f0 , f1 , f2 , . . .) ∈ W ). Show that W is a subspace of V
and find a basis for W . [Hint: How many of the entries of a sequence (x0 , x1 , x2 , . . .) in W
do you need to know before you can determine the entire sequence? Use these entries as your
coordinates.] Write out the first 5 terms of each entry in your basis. What is dim(W )?
(c) Find all geometric sequences of the form (1, r, r2 , . . .) in W . [Hint: What does the first equation
x2 = x0 + x1 tell you about r?] Show that these sequences form a basis for W . [If you have
the right number, you just need to show they are linearly independent] Expect to encounter
irrational numbers in this problem.
(d) Write (f0 , f1 , f2 , . . .) as a linear combination of the geometric sequences you found in (c). Use
this answer to find a closed form formula for fn .1
√ !n
1
1+ 5
(e) Using your answer to part (d), show that fn is the closest integer to √
for all
2
5
n = 0, 1, 2, . . .. Using a calculator, find f50 .
fn+1
(f) Find lim
.
n→∞ fn
1
This is know as Binet’s formula.
3