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Question Number 61850 Answers: 1 Comments: 8

Find all integer solution(s): 615+x^2 =2^y

$${Find}\:{all}\:{integer}\:{solution}\left({s}\right): \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{615}+\boldsymbol{{x}}^{\mathrm{2}} =\mathrm{2}^{\boldsymbol{{y}}} \\ $$

Question Number 61843 Answers: 0 Comments: 3

let V be a vector space and let H and K be subspace of V. show that , H+K={x:x=h+k, where h∈H and k∈K} is a subspace of V.

$$\boldsymbol{{let}}\:\boldsymbol{{V}}\:\:\:\boldsymbol{{be}}\:\boldsymbol{{a}}\:\boldsymbol{{vector}}\:\boldsymbol{{space}}\:\boldsymbol{{and}}\:\boldsymbol{{let}}\:\boldsymbol{{H}}\:\boldsymbol{{and}}\:\boldsymbol{{K}}\:\boldsymbol{{be}}\: \\ $$$$\boldsymbol{{subspace}}\:\boldsymbol{{of}}\:\boldsymbol{{V}}.\:\boldsymbol{{show}}\:\boldsymbol{{that}}\:, \\ $$$${H}+{K}=\left\{\boldsymbol{{x}}:\boldsymbol{{x}}=\boldsymbol{{h}}+\boldsymbol{{k}},\:\boldsymbol{{where}}\:\boldsymbol{{h}}\in{H}\:\boldsymbol{{and}}\:\:\boldsymbol{{k}}\in{K}\right\}\:\boldsymbol{{is}}\:\:\boldsymbol{{a}}\:\boldsymbol{{subspace}}\:\boldsymbol{{of}}\:\boldsymbol{{V}}.\: \\ $$

Question Number 61842 Answers: 0 Comments: 1

consider the space Pn with H={f:f⊂Pn and ∫_0 ^1 f(x)∂x=0} . Show that H is a SUBSPACE of Pn.

$$\boldsymbol{{consider}}\:\boldsymbol{{the}}\:\boldsymbol{{space}}\:\boldsymbol{{P}}{n}\:\boldsymbol{{with}}\:\boldsymbol{{H}}=\left\{{f}:{f}\subset{Pn}\:\boldsymbol{{and}}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right)\partial{x}=\mathrm{0}\right\}\:.\:{S}\boldsymbol{{how}}\:\boldsymbol{{that}}\:\boldsymbol{{H}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{S}}{UBSPACE}\:{of}\:{Pn}. \\ $$

Question Number 61840 Answers: 1 Comments: 0

consider the triple of real numbers (x,y,z) defined by the addittion (x,y,z)+(x′,y′,z′)=(x+x′,y+y′,z+z′) and scalar multiplication by 𝛂(x,y,z)=(0,0,0). Show that all axioms for a vector space are satisfied except axiom 8.

$$\boldsymbol{{consider}}\:\boldsymbol{{the}}\:\boldsymbol{{triple}}\:\boldsymbol{{of}}\:\boldsymbol{{real}}\:\boldsymbol{{numbers}}\:\left(\boldsymbol{{x}},{y},{z}\right) \\ $$$${defined}\:{by}\:{the}\:{addittion}\:\left(\boldsymbol{{x}},{y},{z}\right)+\left({x}',{y}',{z}'\right)=\left({x}+{x}',{y}+{y}',{z}+{z}'\right) \\ $$$$\boldsymbol{{and}}\:\boldsymbol{{scalar}}\:\boldsymbol{{multiplication}}\:\boldsymbol{{by}}\:\:\:\boldsymbol{\alpha}\left({x},{y},{z}\right)=\left(\mathrm{0},\mathrm{0},\mathrm{0}\right).\: \\ $$$$\boldsymbol{{S}}{how}\:{that}\:{all}\:{axioms}\:{for}\:{a}\:{vector}\:{space}\:{are}\:{satisfied}\:{except}\:{axiom}\:\mathrm{8}. \\ $$

Question Number 61855 Answers: 1 Comments: 0

∫_(0 ) ^1 ((3x^3 −x^2 +2x−4)/(√(x^2 −3x+2))) dx

$$\int_{\mathrm{0}\:} ^{\mathrm{1}} \frac{\mathrm{3}{x}^{\mathrm{3}} −{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{4}}{\sqrt{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}}\:{dx} \\ $$

Question Number 61835 Answers: 0 Comments: 1

If α = Cis(2π/7) and f(x) = A_0 + Σ_(n=1) ^(14) A_n x^n Then prove that Σ_(α=0) ^6 f(α^n x)= 7(A_0 +A_7 x^7 +A_(14) x^(14) ) where Cisθ = Cosθ + iSinθ

$$\mathrm{If}\:\alpha\:=\:\mathrm{Cis}\left(\mathrm{2}\pi/\mathrm{7}\right)\:\mathrm{and}\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{A}_{\mathrm{0}} \:+\:\sum_{\mathrm{n}=\mathrm{1}} ^{\mathrm{14}} \mathrm{A}_{\mathrm{n}} \mathrm{x}^{\mathrm{n}} \: \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}\:\sum_{\alpha=\mathrm{0}} ^{\mathrm{6}} \mathrm{f}\left(\alpha^{\mathrm{n}} \mathrm{x}\right)=\:\mathrm{7}\left(\mathrm{A}_{\mathrm{0}} +\mathrm{A}_{\mathrm{7}} \mathrm{x}^{\mathrm{7}} +\mathrm{A}_{\mathrm{14}} \mathrm{x}^{\mathrm{14}} \right) \\ $$$$\mathrm{where}\:\mathrm{Cis}\theta\:=\:\mathrm{Cos}\theta\:+\:\mathrm{iSin}\theta \\ $$

Question Number 61834 Answers: 0 Comments: 3