2. (a) Let W be a vector space with inner product (,) and the norm ||x|| = √(x,x).Prove that, for u, w € W,||u+w||² + ||uw||² = 2||u||²+2||w||².Let V be a vector space. A norm is a function v→ ||v|| that satisfies|v|| ≥ 0, for v € V.P1:P2:||v|| = 0 if and only if v = 0.P3:||rv|| = |r|||v||, for v € V and r ER.P4: ||u+v||≤|u|| + ||v||, for v € V.(b) Definition:For example, for= √+ is called the 2-norm on R².The 1-norm and the infinity norm on R² are defined asAL·IOL·i. Show that the 1-norm satisfies properties P1 to P4.ii. Show that the 1-norm does not satisfy Equation (1).|³x| + |¹x| =and= max(₁, ₂), respectively.iii. Show that the infinity-norm satisfies properties P1 to P4.iv. Show that the infinity norm does not satisfy Equation (1).(That is, we cannot write ||x|1₁ nor ||x|| as √(x,x), for any inner product on R².)

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Answered: 2. (a) Let W be a vector space with…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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18. Let V = {(a1, a2): a1, a2 € R}. For (a₁, a2), (b₁,b₂2) € V and c € R, define (a1, a2) + (b1,b2) = (a1 + 2b1, a2 + 3b2) and c(a1, a2) = (ca₁, ca₂). Is V a vector space over R with these operations? Justify your answer.
TASK 23
2. Let E be a vector space. A function p: E ->e is culled a seminorm it Pirx)=1ripx) and puty) spultpg) for all x,y e E and reIR. Give on example a 2. IR which is not a norM. nonzero seminorm
5. Let V be a vector space, and let x1, X2, ... , Xk E V. 5(a) Prove that if either x1 = 0 or x;+1 E span{x1,...,x;} for some j = 1,2,... , k – 1, then x1, X2,.. dependent. %3D
Sketch each of the following sets and decide if they are open sets. Give brief reasons for your decision. (a) The open interval (-1, 1) in the normed vector space (R,| |). (b) The closed interval |-1, 1] in R. (c) The interval [0, 0) in R. (d) The line {(2x, x): a E R} in R². (e) The square (0, 1] x [0, 1) in R2. (f) The set {(0,0, 0)}U (0, 1) x (0, 1) x (0, 1) in R.
2. Let V, W be subsets of R“ defined the following way: a а 2а + b a+b V = a,b ER W : a,b €R, a<0
3. Which of the following pairs of vector spaces are isomorphic? Justify your answers. F³ and P3 (F). F4 and P3(F). (a) (b) (c) (d) M2x2 (R) and P3(R). V = {A € M₂x2 (R) : tr(A) = 0} and R4.
Q4: Let X R³ be a vector space F and||||= ₁² +₂1² +131² V (11, 12, 13) E R³ show that whether (R³, || ||) is normed space or not?
(CS) is called the Cauchy-Schwarz inequality and (PI) is called the parallelogram identity.
Definition 28. Let V be a vector space over F (here F= R, C). Then the function (•₁•): V × V → F is said to be an inner product on V if the following conditions are satisfied: (x, x) is real and (x,x) ≥ 0 for all (x,x) = 0 if and only if x € 0₂. (i) (ii) (iii) (x, ay) = a(x, y) for all x, y V (iv) (x, y+z) = (x, y) + (x, z) for all (v) (x, y) = (y,x) for all x, y V. (a) (.,.): R¹ × R¹ → R is defined as xεν. (x, y) = x¹y, and for all x,y,z € V. Example: Verify that each of the following mappings define a real inner product. a EF. for all x, y ER".
(1) Consider v = (v₁) € R" (v; are the components of v). (la) standard (₂) vector norm ||v|| = √vTv (1b) the maximum norm ||v||max = max; [v₁], where v; are the components of v = (v₁) (1c) the ₁ vector norm ||v||₁=;|v₁| Compute these norms for v = √2
1. Let f : R" → R" be a function with the property that for any v, w E R" it holds that f(v) · f(w) = V· W. I.e. f preserves the inner product on the vector space R". Show that f is a linear transfor- mation.

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