If x and y are two n-vectors, then prove that 1 1 (a) x'y< I,l»,- +=1 (Holder incquality) P 9 (b) xy, < ,, (Schwarz inequality).
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- Suppose that the 100-vector x gives the age distribution of some population, with xi the number of people of age i-1, fori=1,...,100. The total number of people with age between 5 and 18 (inclusive) is given by x6+x7+..+x18+x19. We can express this as s^Tx, where s is the vector with entries one for i=6,..,19 and zero otherwise. Generate a randomized x with entries between 0 and 100. HINT: rand(n) generates an n-vector with values between 0 and 1. how can you scale its entries to be between 0 and 100?\(\) Choose x such that the following two vectors are parallel u= and U = O None of the others O (1) 6 (1) -3 16, x, x, O O (1) 2 O \(1) 3Let S = {v1, v2, v3} be a set of linearly independent vectors in R^3. (a) Are the vectors in the set T1 = {v1, v1 + v3, v3} linearly independent? Show how you arrived at your answer. (b) Are the vectors in the set T2 = {v1 − v3, 3v1 + v2, v2 + 5v3} linearly independent? Show how you arrived at your answer.
- 3. If x and y are two n-vectors, then prove that 1 1 (a) |x²y|≤|x, y, › + =1 (Holder inequality) P 9 (b) |xy²|₂ ≤|x]|₂||y||₂ (Schwarz inequality).The dot product of two vectors X1 Yi X2 y2 and y = X = Xn Yn in R" is defined by x· y = x1yı + x2Y2 + perpendicular if x· y = 0. Any vector in R' perpendicular to X • + xnYn. The vectors x and y are called ... 3 6. can be written in the form s+ t.2 Let =? and P vectors, 2, in S panEu} and orthogonal to ū Write P as the sum of two orthogonal
- Define' a vector to contain 2 components: (a,b) where a a rational number, b is an irrational number. Let S? be a set that contains such vectors, ie. s*={ (a,b)| for all rational numbers a, irrational numbers b}. For example, (3.14,71) is in S², but (e,71) is not in S² – 1 & e are well known math constants. Is S a vector space over the integers?01 3.) Verify the Cauchy - Sahorz inequality and the triangle inequality for the vectors. 3. x=(2, 0₁-1) (4,0,-2) ¡y = 4.) x=(7,0,2, 6), y =(3, 8, 4, 1) 5.) 6.) x - (1₁-1, 1,-1, 1), y=(3, 0, 0, 0, 2) = (1,0,0,1), y=(-1,0,0,1).,Let x and y be any two vectors in R^ 3 . a. Prove that the vectors ||y||x + ||x||y and ||y||x - ||x||y are orthogonal. b. Show that ||y||x + ||x||y bisects the angle between x and y.
- Find three numbers a1, a2, az to normalize the orthogonal set {-E-F-} 1 2 a1 1 ,a3 3 1 in R3. , Az = azShow that the vectors X1 = 5e3t and (3)-* 2et X2 = Tet form a linearly independent set.Let v1 vector=[−4 h] and v2 vector =[−10 20] (check image for reference) Then span(v1 vector,v2 vector)=R^2 as long as h≠ (If there are no values of hh that make the set fail to span R^2, enter "NONE". If the set fails to span R^2 for all values of hh enter "ALL".)