5

# Consider the basis S = {1,1,22 in the vector space Pz, the set of all polynomials of degree at most 2, where Pz is a subspace of L?[0,1]. (a) Show that S = {1,2,22}...

## Question

###### Consider the basis S = {1,1,22 in the vector space Pz, the set of all polynomials of degree at most 2, where Pz is a subspace of L?[0,1]. (a) Show that S = {1,2,22} is not an orthogonal set in L?[0,1]. (b) Use the Gram-Schmidt algorithm to generate an orthonormal basis E = {e1,e2, e3_ from S Use E to find the polynomial p = p(z) of degree at most 2 that best approximates 1/2 < x < 1 f() = 0 0 < r < 1/2 (i.e , find the (orthogonal) projection of f onto the space Pz spanned by E)_ (d S

Consider the basis S = {1,1,22 in the vector space Pz, the set of all polynomials of degree at most 2, where Pz is a subspace of L?[0,1]. (a) Show that S = {1,2,22} is not an orthogonal set in L?[0,1]. (b) Use the Gram-Schmidt algorithm to generate an orthonormal basis E = {e1,e2, e3_ from S Use E to find the polynomial p = p(z) of degree at most 2 that best approximates 1/2 < x < 1 f() = 0 0 < r < 1/2 (i.e , find the (orthogonal) projection of f onto the space Pz spanned by E)_ (d Sketch p and f on the same set of axes.

#### Similar Solved Questions

##### ExercisesThe fundamental property of the exponential function is the addition formula, which tells us that exp maps sums to products, that is, eA+B = eAeB . However; we are about to generalize the exponential function to objects that do not enjoy all the algebraic properties of real or complex numbers, so itis important to investigate whether the equation eA+B = eAeB still holds. The answer is that it does, provided AB = BA. We assume that X X2 =1+ + + where 1 is the identity object 1! 2!
Exercises The fundamental property of the exponential function is the addition formula, which tells us that exp maps sums to products, that is, eA+B = eAeB . However; we are about to generalize the exponential function to objects that do not enjoy all the algebraic properties of real or complex numb...
##### (25) y + % =1 (27) cy' +y = ry (t+6y (29) 2y' + ytanâ‚¬ COs I(26) y + Jy = ety' (28) y +Zy = ~Ty} cos1 (30) wy' +y =y12Inz
(25) y + % =1 (27) cy' +y = ry (t+6y (29) 2y' + ytanâ‚¬ COs I (26) y + Jy = ety' (28) y +Zy = ~Ty} cos1 (30) wy' +y =y12Inz...
##### Numocr of vicwers of (cicvision scrics Introducco Scvcral Ycars 0g0 N(t) 2t)2/3 (1 3 [Jppro imatccthe functionWhere N(t) (measured Millions) denotes the number weekit viewers of the series of week 16 (Round ~cur answens- decimam Place ) week million/weekthe end of the tth week: Find the rate Increasethe weekly dudiencezthe endWeekand the endweek 16milliontweeHow mhjny Viewcrs Were cherewcckwcek 267wcckrnionWeek 26mlllonNeed Help?-13 pointsTANAPMATH7 9.6.072.My NotesAsk Your TeacherThe population
numocr of vicwers of (cicvision scrics Introducco Scvcral Ycars 0g0 N(t) 2t)2/3 (1 3 [ Jppro imatcc the function Where N(t) (measured Millions) denotes the number weekit viewers of the series of week 16 (Round ~cur answens- decimam Place ) week million/week the end of the tth week: Find the rate Inc...
##### 3 3 1 3 Tli 6 1 3 I 2 Ii 3 1 U 2 5 ? 8 0 8 8 8 | } 1 I 8 1 3 1 8 1 3 B1E E 3 L | 1 U [ L I 1 1 1 1 1 1 ( 1 1 1 8
3 3 1 3 Tli 6 1 3 I 2 Ii 3 1 U 2 5 ? 8 0 8 8 8 | } 1 I 8 1 3 1 8 1 3 B1E E 3 L | 1 U [ L I 1 1 1 1 1 1 ( 1 1 1 8...
##### Parametrize and sketch the surface % = :? + y? , 0 < 2 < 2b) Find the area of this surfacec) Find the fux across the surface i.e [ Js F ndo where F is given by F = yli+rzj-k and n is the unit normal pointing away from the 2 aris_ Do the surface integral
Parametrize and sketch the surface % = :? + y? , 0 < 2 < 2 b) Find the area of this surface c) Find the fux across the surface i.e [ Js F ndo where F is given by F = yli+rzj-k and n is the unit normal pointing away from the 2 aris_ Do the surface integral...
##### In the circuit shown, Ri-Rz-R, and 82 = &- The power dissipated in R1 is 20 W: Does battery 81 supply or absorb energy; what is the rate of energy 81 exchanges with the circuit? Assume that the batteries are ideal: RzeR AM MMM 6j Zr; Er R 8z MM RNone of themAbsorb 440 J/SSupply 48 WAbsorb 220 J/sSupply 110 WAbsorb 1 WSupply 440 J/SAbsorb 24 J/s
In the circuit shown, Ri-Rz-R, and 82 = &- The power dissipated in R1 is 20 W: Does battery 81 supply or absorb energy; what is the rate of energy 81 exchanges with the circuit? Assume that the batteries are ideal: RzeR AM MMM 6j Zr; Er R 8z MM R None of them Absorb 440 J/S Supply 48 W Absorb 22...
##### 8.48cdn fbsbx comDone0 - b 20"0 0,02 0.200" 0 7Nine raised t0 four iS equal t0 6561 b 36 2.360 d 656 Find the discount and sale price ofan item that sell for php 298.95 with an 18% discount Php 53.81 b Php 53.00 Php 53.85 Pbp 53.1 9.It is the money that is paid for the use of money over a period of time. Discount Interest Commission Integers 10.Find the integers on php 3000 for 2 years at rate of 5% per ycar 400 b. 300 500 4.550IL Auswer the following questions; 1-3 (-4) 2.7 +
8.48 cdn fbsbx com Done 0 - b 20"0 0,02 0.200" 0 7Nine raised t0 four iS equal t0 6561 b 36 2.360 d 656 Find the discount and sale price ofan item that sell for php 298.95 with an 18% discount Php 53.81 b Php 53.00 Php 53.85 Pbp 53.1 9.It is the money that is paid for the use of money over...
##### [email protected]) Cive a possible set of quantum numbers for an = level. (2 points) electron in & Pe orbital in the 4u cnergyb) Give possible set of level: (2 points) quantum numbers for an electron . in a orbital in the 5t energy
[email protected]) Cive a possible set of quantum numbers for an = level. (2 points) electron in & Pe orbital in the 4u cnergy b) Give possible set of level: (2 points) quantum numbers for an electron . in a orbital in the 5t energy...
##### Asimple random sample from nonulabon wrh norMa disiribulion cggtiytemderaiures has conclude Ihat Ihe populalion standard deviaiion ess (nan 80" F?99 1020 68* Constnid 909 confidence interva esiimate of ine standard davialion ol body emnerahime Dla headhv hiMansiealeCuckIhe IconMeeIne IahleCISouare cuical ualues<<U"F (Rounn hvo decimal placesDENded `IL saleconchide Inal Ine ccnulalon etandard deviaionless than 80 F?amais cnncnannol Sale bucuuseconlioencu intuvalThis conchsonsale b
Asimple random sample from nonulabon wrh norMa disiribulion cggtiytemderaiures has conclude Ihat Ihe populalion standard deviaiion ess (nan 80" F? 99 102 0 68* Constnid 909 confidence interva esiimate of ine standard davialion ol body emnerahime Dla headhv hiMans ieale CuckIhe Icon MeeIne Iahle...
##### Qivesulotn'potassium flucride KF; that has concentration; [(m/m)%] ofa soluticn of Calculate the mass percent = 8.5 & of KF dissolved in 93.0 & of water:8.470 1151125all inconectQuestion 96 ptsWhich of the following is are general properties of a solution?Ca mixture in which dissolved solids substance moleculat or Ionic break down into individual molecules or ions depending whether the Aand C} are trueBJa mixture of variable composition Nahonogereous ' mixture of two or more su
Qivesulotn 'potassium flucride KF; that has concentration; [(m/m)%] ofa soluticn of Calculate the mass percent = 8.5 & of KF dissolved in 93.0 & of water: 8.47 0 115 1125 all inconect Question 9 6 pts Which of the following is are general properties of a solution? Ca mixture in which di...
##### Suppose 500. mL flask is filled with 1.7 mol of Oz and 0.40 mol of SO; This reaction becomes possible:2SO_(g)+02(g) 2S03 (g)Complete the table below, that it lists the initial molarity of each compound the change in molarity of each compound due to the reaction, and the equilibrium molarity of each compound after the reaction has come t0 equilibrium:Use x to stand for the unknown change in the molarity of Oz: You can leave out the Msymbol for molarity:S0_S0_initialchangeequilibrium
Suppose 500. mL flask is filled with 1.7 mol of Oz and 0.40 mol of SO; This reaction becomes possible: 2SO_(g)+02(g) 2S03 (g) Complete the table below, that it lists the initial molarity of each compound the change in molarity of each compound due to the reaction, and the equilibrium molarity of eac...
##### The accompanying figure shows the graphs of $y=f(x)$ and $y=\int_{0}^{x} f(t) d t .$ Determine which graph is which, and explain your reasoning. (FIGURE CAN'T COPY)
The accompanying figure shows the graphs of $y=f(x)$ and $y=\int_{0}^{x} f(t) d t .$ Determine which graph is which, and explain your reasoning. (FIGURE CAN'T COPY)...
##### Ccnvert thc 9s(0) Qpola equation1 DETAILS Jeiod Osprecalci of the 1 3Veisa)puz 0(Usequimoio} Mecessan: Xand
Ccnvert thc 9s(0) Qpola equation 1 DETAILS Jeiod Osprecalci of the 1 3 Veisa) puz 0 (Use quimoio} Mecessan: Xand...
##### Let $$\begin{array}{l}{f(x)=-3 x^{3}-2 x^{2}-x+4} \\ {g(x)=x^{3}-x^{2}-5 x-4} \\ {h(x)=-2 x^{3}+5 x^{2}-4 x+1}\end{array}$$ Find the indicated function, function value, or polynomial. $$-2 g(x)-3 h(x)$$
Let $$\begin{array}{l}{f(x)=-3 x^{3}-2 x^{2}-x+4} \\ {g(x)=x^{3}-x^{2}-5 x-4} \\ {h(x)=-2 x^{3}+5 x^{2}-4 x+1}\end{array}$$ Find the indicated function, function value, or polynomial. $$-2 g(x)-3 h(x)$$...
##### Coripkte the foliowving reaction schemes: Somte may take more than ore 5Pg and sonla Inay have several routes to the product. Peints are granted based o the vzlidity of the scheme and efficiency (least amount of steps}:
Coripkte the foliowving reaction schemes: Somte may take more than ore 5Pg and sonla Inay have several routes to the product. Peints are granted based o the vzlidity of the scheme and efficiency (least amount of steps}:...
##### Decomposition rcaction of potassium chlorate shown below .KCIO;KCI 3 0z(a) What is the rate of formation Of potassium chloride and oxygen? What is the rate of decomposition of potassium chlorate? How is the Tate of formation of oxygen related t0 the rale of decomposition of potassium chlorate?The reaction between nitrogen monoxide and ozone t0 fonm nitrogen dioxide and oxygen is shoun below. NO ( 0, (4 NO: () 0161(a) If rate law is r =KNO] [O:] detennine the overall order and unit ofk for the re
Decomposition rcaction of potassium chlorate shown below . KCIO; KCI 3 0z (a) What is the rate of formation Of potassium chloride and oxygen? What is the rate of decomposition of potassium chlorate? How is the Tate of formation of oxygen related t0 the rale of decomposition of potassium chlorate? Th...