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 , then they are linearly independent. 2. Recall the denit
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 be multiplied, and in which order, and compute the prod
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 on R2 where (a, b) (c, d) if and only if a2 + b2 = c2
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. Prove the following equalities (a) A B = (A B ) (A B ).
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 now. (a) The main course will be either beef or sh. The
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 convergence. By properties of convergent sequences (Theorem 1.1.
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 < p1 < xp1 . 2. (a) xn 1 nxn1 = lim = n. x1 x2 x x1 2x 1 l
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 a [0, 2 ) such that v = u() where u() = .
2. Find the c
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 (x, y ) = (x y 2 , 5x). (c) T : R2 R2 , T (x, y ) = (x 1
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 )2
and taking the square root gives the triangle inequal
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) = R S (T (x) = R(S (T (x) and R (S T )(x) = 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 ), then T (x + x , y + y ) = (x + x y y , 5x +5x ) = (x y +
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 any v V we have x v = 0. Show that x V is equivalent t
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
vj
i=1
i vi
=
i=1
i vj vi = i vj vj ,
but vj = 0 and
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
in terms of the dot product.
2. Use the Cauchy Schwarz
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 : Rm Rl and R : Rl Rk are linear maps then (R S ) T = R
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 Use this formula to derive the following relations cos(3)
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 is vacuously true (as (m, M ) = .) Otherwise set x = mi
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 . Choose xn Q. Then f (xn ) = 0 for all n N, and it is
Analysis 1: Exercises 14
1. Let (a(n) be a decreasing sequence of strictly positive numbers with n-th partial sum s(n). (a) By grouping terms in two dierent ways, show that 1 cfw_a(1) + 2 a(2) + + 2n a(2n ) s(2n ) a(1) + 2 a(2) + + 2n1 a(2n1 ) + a(2n ). 2
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 , (a) lim 2 x 1 x x (1 x)m xm (b) lim , n n x 1 (1 x) x 2
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. Prove that f + g is also dierentiable at a, and (f + g ) (a
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 in [2, 0]. (a) x3 x + 3 = 0; (b) x5 + x + 1 = 0. 3. Le
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. Let f and g be continuous at a R. Prove that (a) f + g i