MAS3702 2017-2018 Example Sheet 1 PDF

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Please solve the following problems:Qu, Qu(iii), Qu(vi), Qu, Qu and Qu;hand in your solutions on Monday the 16th of October by 16. The tutorial is on Monday the 9th of October at 11 in ARMB.3.Z.A MAS3702 Linear Analysis (2017) Examples Sheet 1Qu. 1 Letv= (1,−2) andw= (3,2) inR 2. Recall that an inne...

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Please solve the following problems: Qu.1, Qu.3(iii), Qu.3(vi), Qu.4, Qu.13 and Qu.16; hand in your solutions on Monday the 16th of October by 16.00. The tutorial is on Monday the 9th of October at 11.00 in ARMB.3.38.

Z.A.Lykova MAS3702 Linear Analysis (2017) Examples Sheet 1 Qu.1

Let v = (1, −2) and w = (3, 2) in R2 . Recall that an inner product in R2 is < v, w > = v1 w1 + v2 w2 .

Find 3v − w, < w, w >, < v, w > and < 3v − w, w >. Show that < 3v − w, w > = 3 < v, w > − < w, w > . 10 marks Qu.2 Let f (t) = −t and g(t) = it2 in C[0, 1], where i2 = −1. Find 2if + g, < f, g > and < g, 2if + g >, where < f, g > = Show that

Z 1 0

f (x)g(x) dx

< g, 2if + g > = −2i < g, f > + < g, g > . Qu.3 Show that the following sets are linear subspaces: (i) {(x, y, z) ∈ R3 : x + y + z = 0} of R3 ; (ii) {A ∈ M2 (R): A = AT } of M2 (R), where AT denotes the transpose matrix of A; ∞ ∈ CN : x2n+1 = 0 for all n ∈ N} of CN ; (iii) {x = (xn )n=1

15 marks (iv) P2 [0, 1] = {f ∈ C [0, 1]: f is a polynomial of degree two or less} of C[0, 1]; (v) P [0, 1] = {f ∈ C [0, 1]: f

is a polynomial } of C [0, 1];

(vi) W = {f ∈ C[0, 1]: f (0) = 0} of C[0, 1]. 15 marks Give an example of a non-zero element of each of the subspaces above. ∞ ∈ CN : supn∈N |xn | ≤ 1} is not a linear subspace Qu.4 Show that W = {x = (xn )n=1 N of C . 20 marks Qu.5 Qu.6

Show that W = {z ∈ C: Rez ≥ 0} is not a linear subspace of C. We denote by ln∞ the normed space (Cn , k · k∞ ), where kvk∞ =

max |vj | for v = (v1 , . . . , vn ).

j=1,... ,n

Find k(1, 1/2, 1/3, . . . , 1/n)k∞ and k(1, 2, 3, . . . , n)k∞ . Qu.7

The 2-norm on C[0, 1] is defined by kf k2 =

Z 1 0

2

1/2

|f (x)| dx

. Find kf k2

when f (x) = xe−2x . Qu.8 Let v = (1, i, i2 , . . . , in ) and w = (1, (1 + i), (1 + i)2 , . . . , (1 + i)n ) in Cn+1 . Find kvk2 , < v, w >. [Hint: G.P.] 1

2

Qu.9

We denote by lnp the normed space (Cn , k · kp ), where kvkp =

 n X 

|vj |

p

j=1

1/p 

for v = (v1 , . . . , vn )



and 1 ≤ p < ∞. What is the definition of kvkp for v ∈ C3 ? Find k(−1, 10, 2)k1 , k(−1, 10, 2)k2 , k(−1, 10, 2)k8 and k(−1, 10, 2)k64 to 4 decimal places. Why do you think k· k∞ is so called? Qu.10

Let v be a vector in C3 . Show that kvk∞ ≤ kvk1 ≤ 3 kvk∞ .

Give examples of two vectors to show that equality might hold on either side. Qu.11 We denote by C pn the normed space (Mn (C), k · kp ), where  n X

kAkp = 

1/p 

p

j=1

|sj (A)| 

,

1 ≤ p < ∞ and sj (A), j = 1, . . . , n, are the singular values ( or s-numbers) of A. Recall that sj (A), j = 1, . . . , n, are the square roots of the eigenvalues of the matrix A∗ A. (It is true, but not at all obvious, that k · kp is a norm on Mn (C).) What is the definition of kAk∞ for A ∈ Mn (C)? Let A ∈ M2 (C) be given by A=

"

1 2 3 4

#

Find the singular values of A, to 4 decimal places. Find approximate values for kAk1 , kAk2 and kAk∞ . Qu.12 Verify that k · k1 is a norm on Cn , where kvk1 =

n X

|vj | for v = (v1 , . . . , vn ).

j=1

We denote by l1n the normed space (Cn , k · k1 ). Qu.13 We denote by c0 the space of complex sequences which tend to zero, i.e., c0 = {v = (v1 , v2 , . . . , vn , . . . ): vi ∈ C, vi → 0 as i → ∞}. Verify that k · k∞ is a norm on c0 , where kvk∞ = max |vi | for v = (v1 , . . . , vn , . . . ). i∈N

20 marks Qu.14

Verify that kxk2 =

 n X 

|xj |

j=1

1/2 

2



,

where x = (x1 , . . . , xn ), xj ∈ C, defines k · k2 as a norm on Cn . Find kx0 k2 , where x0 = (1, i, 1, i, . . . ). ∞ Qu.15 ℓ∞ denotes the complex vector space of all bounded sequences x = (xn )n=1 of complex numbers, with componentwise addition and scalar multiplication. Verify that k · k∞ is a norm on ℓ∞ , where kxk∞ = sup |xn |. n∈N

3

Qu.16 Let c0 and (ℓ∞ , k· k∞ ) be as in Qu.13 and Qu.15 respectively. Show that c0 is a linear subspace of ℓ∞ . 20 marks

Qu.17

Prove the Cauchy-Schwarz inequality: n X

a k bk ≤ (

n n X X 2 12 2 12 ) )

ak

k=1

k=1

(

bk

k=1

for all ak , bk ∈ R, ak , bk ≥ 0, k = 1, 2, . . . , n, where n ∈ N. Hint: Consider the quadratic

Pn

k=1 (ak

+ tbk )2 in t....