Linear Algebra & Geometry: solutions 4
1. We consider v+w
2
= (v + w) (v + w) = v v + 2v w + w w = v 2 + w 2 + 2v w w and obtain w
Now we use Cauchy Schwarz v w |v w| v v+w
2
v 2+ w 2+2 v = ( v + w )
Linear Algebra & Geometry: Sheet 1
Set on Friday, October 9: Question 1 (a) (b) (c), Question 3, Question 6 (a) (d) and Question 7
1. Sketch the following vectors in R2 and compute their norm v (a) v1
Linear Algebra & Geometry: Sheet 5
Set on Friday, November 6: Questions 1, 2, 3, and 4
1. Show that if the k vectors v1 , v2 vk Rn are all non-zero and mutually orthogonal, i.e., vi vj = 0 if i = j ,
Linear Algebra & Geometry: Sheet 9
Set on Friday, Dec 4: Questions 1, 2, 3 and 4
1. Consider the following matrices A= 00 12 B= 1 1 C= 4 2 0 1 D = 2 0 31 E= 2 01 1 3 0
and determine which of them can
Analysis: Exercises 3
1. [in F.T.A.] (a) Let f : X Y be a function. Dene the relation R on X 2 by R = cfw_(x, y ) X X | f (x) = f (y ). Prove that R is an equivalence relation. (b) Let be the relation
Analysis 1: Exercises 2
1. Let A, B be sets from a universe. Prove the following properties of the complement. (a) (Ac )c = A. (b) (A B )c = Ac B c . (c) (A B )c = Ac B c . 2. Let A, B and C be sets.
Analysis 1: Exercises 1
1. Identify the premises and conclusions of the following deductive arguments and analyse their logical form. Do you think the reasoning is valid? Just use your intuition for n
ANALYSIS EXERCISE 15SOLUTIONS
1. Set Am := 1 an , A := m 1 ( an + bn ). Then
m
1
an , Bm :=
m 1 bn ,
B :=
1 bn
and Cm :=
Cm = A m + B m . Also, Am A as m and Bm B as m from the denition of convergen
ANALYSIS EXERCISE 13SOLUTIONS
1. For the function f (t) = tp write the Mean value theorem f (x) f (y ) = f ( )(x y ), where x > > y > 0, i.e. xp y p = p p1 (x y ). It is enough to notice that y p1 < p
Linear Algebra & Geometry: Sheet 2
Set on Friday, October 16: Questions 1, 2, 4 (a)(c)(e), 5 (i), 6
1. Consider the following vectors: ( a) 1 1 (b) 0 5 cos sin (c) 3 4 ,
nd for each of them a > 0 and
Linear Algebra & Geometry: Sheet 7
Set on Friday, November 20: Questions 1, 2, and 3
1. Determine which of the following maps are linear maps (a) T : R2 R2 , T (x, y ) = (x y, 5x). (b) T : R2 R2 , T (
Linear Algebra & Geometry: Solutions to sheet 8
1. (a) Let x Rn , then R T (x) = R(T (x), S T (x) = S (T (x) and (R + S ) T (x) = (R + S )(T (x) = R(T (x) + S (T (x). (b) Let x Rn , then (R S ) T (x)
Linear Algebra & Geometry: Solutions to Sheet 7
1. We have to check that (i) T (x + y) = T (x) + T (y) and (ii) T (x) = T (x). (a) T (x, y ) = (x y, 5x) is linear. (i) let x = (x, y ), y = (x , y ), t
Linear Algebra & Geometry: Sheet 6
Set on Friday, November 12: Questions 1, 2, and 3
1. Let V Rn be a linear subspace and u1 , u2 , , uk an orthonormal basis of V . (a) For x Rn we say that x V if for
Linear Algebra & Geometry: Solutions to sheet 5
1. We have to show that if
k
i vi = 0
i=1
then 1 = 2 = = k = 0. Let us take the dot product of the sum with vj , then since vj vi = 0 if i = j we nd
k k
Linear Algebra & Geometry: Sheet 4
Set on Friday, October 30: Questions 2, 3, 4, and 6
1. Use the Cauchy Schwarz inequality |v w| v w to derive the triangle inequality
v+w v + w Hint: express v + w
2
Linear Algebra & Geometry: Sheet 8
Set on Friday, November 27: Questions 1, 2, 3 and 5
1. Show that (a) if T : Rn Rm and R, S : Rm Rk are linear maps, then (R + S ) T = R T + S T (b) if T : Rn Rm , S
Linear Algebra & Geometry: Sheet 3
Set on Friday, October 23: Questions 1, 2, 3,4
1. Use the relation ei = cos + i sin to prove De Moivre s Theorem : for any n N cos(n) + i sin(n) = (cos + i sin )n Us
ANALYSIS EXERCISE 12SOLUTIONS
1. Since the function f is continuous on [a, b] there are cfw_x1 , x2 [a, b] such that f (x1 ) = m, f (x2 ) = M . If m = M then f is a constant function, and the result
ANALYSIS EXERCISE 11SOLUTIONS
1. By the denition of continuity for every x0 [a, b]
xx0
lim f (x) = f (x0 ).
Using the Heine denition one can compute the above limit along the sequences xn x0 , xn = x0
Analysis 1: Exercises 13
1. Let p > 1 and x > y > 0. Use the Mean Value Theorem to prove the inequality py p1 (x y ) xp y p pxp1 (x y ). 2. Let m, n N. Use lHopitals rule to compute the limits xn 1 ,
Analysis 1: Exercises 12
1. Let f be a continuous function on [a, b]. Denote M := max f, m := min f.
[a,b] [a,b]
Prove that (c (m, M ) x0 (a, b) f (x0 ) = c .
2. Let f and g be dierentiable at a. Prov
Analysis 1: Exercises 11
1. Let f be continuous on [a, b]. Suppose that f (x) = 0 for all rational x [a, b]. Prove that f (x) = 0 for all x [a, b]. 2. Prove that the following equations have solutions
Analysis 1: Exercises 10
1. (a) Let f be continuous at a R. Prove that |f | is continuous at a. (b) Let f and g be continuous at a R. Prove that maxcfw_f, g and mincfw_f, g are continuous at a. 2. L
Analysis 1: Exercises 9
1. (a) Prove that
xa xa
lim f (x) = lim f (a + h).
h0 x a
(b) Prove that lim f (x) = b if and only if lim f (x) b = 0. 2. Prove the following two theorems. Theorem 1. Let lim f