1

Let $V$ be a vector space over $K$. A mapping $f: \widehat{V \times V \times \ldots \times V} \rightarrow K$ is called a multilinear (or $m$ -linear) form on $V$ if...

Question

Let $V$ be a vector space over $K$. A mapping $f: \widehat{V \times V \times \ldots \times V} \rightarrow K$ is called a multilinear (or $m$ -linear) form on $V$ if $f$ is linear in each variable; that is, for $i=1, \ldots, m$ \[ f(\ldots, \widehat{a u+b v}, \ldots)=a f(\ldots, \hat{u}, \ldots)+b f(\ldots, \hat{v}, \ldots) \] where $\widehat{\longrightarrow}$ denotes the $i$ th element, and other elements are held fixed. An $m$ -linear form $f$ is said to be alternating if $f\left(v_{1}, \ldots

Let $V$ be a vector space over $K$. A mapping $f: \widehat{V \times V \times \ldots \times V} \rightarrow K$ is called a multilinear (or $m$ -linear) form on $V$ if $f$ is linear in each variable; that is, for $i=1, \ldots, m$ \[ f(\ldots, \widehat{a u+b v}, \ldots)=a f(\ldots, \hat{u}, \ldots)+b f(\ldots, \hat{v}, \ldots) \] where $\widehat{\longrightarrow}$ denotes the $i$ th element, and other elements are held fixed. An $m$ -linear form $f$ is said to be alternating if $f\left(v_{1}, \ldots v_{m}\right)=0$ whenever $v_{i}=v_{j}$ for $i \neq j .$ Prove the following: (a) The set $B_{m}(V)$ of $m$ -linear forms on $V$ is a subspace of the vector space of functions from $V \times V \times \cdots \times V$ into $K$ (b) The set $A_{m}(V)$ of alternating $m$ -linear forms on $V$ is a subspace of $B_{m}(V)$ Remark 1: If $m=2,$ then we obtain the space $B(V)$ investigated in this chapter. Remark 2: If $V=K^{m}$, then the determinant function is an alternating $m$ -linear form on $V$



Answers

Let $F$ be a fixed $3 \times 2$ matrix, and let $H$ be the set of all matrices $A$ in $M_{2 \times 4}$ with the property that $F A=0$ (the zero matrix in $M_{3 \times 4} ) .$ Determine if $H$ is a subspace of $M_{2 \times 4}$ .


Similar Solved Questions

5 answers
Time Ielt _.Minimum vaiue101.27:13Need Help? Tulk lo 4Tutolpoints SCalcB 14.8.031.My NotesUse Lagrange multipliers to find the shortest distance from the point (2, 0, -7) to the plane X+ + z =Need Help? Tello torSubrat AnswerSavu ProgrussPraclicu Anolhur VurslonI/1 poIntsPreviq u; Answens SCake 14.8.035.My NotesUse Lagrange multipliers to find three positive numbers whose sum is 180
time Ielt _. Minimum vaiue 10 1.27:13 Need Help? Tulk lo 4Tutol points SCalcB 14.8.031. My Notes Use Lagrange multipliers to find the shortest distance from the point (2, 0, -7) to the plane X+ + z = Need Help? Tello tor Subrat Answer Savu Progruss Praclicu Anolhur Vurslon I/1 poInts Previq u; Ans...
5 answers
Reverse the order of integration; and then integrate (Or just integrate, to receive partial credit:) [" rydxdy
Reverse the order of integration; and then integrate (Or just integrate, to receive partial credit:) [" rydxdy...
5 answers
<3 > A circuit has in series an electromotive force given by E(t)= 12 (v); a resistor of 2 ohms, an inductor of H and a capacitor of 0.2 F: If the initial charge is 0 and the initial current is 0, find the charge at any time &
<3 > A circuit has in series an electromotive force given by E(t)= 12 (v); a resistor of 2 ohms, an inductor of H and a capacitor of 0.2 F: If the initial charge is 0 and the initial current is 0, find the charge at any time &...
5 answers
1.6 CYLINDERS AND QUADRIC SURFACES9.Find the traces of the following surfaces in the planes parallel t0 the coordinate planes and identity each the surface. (a) ? v - ?=0; (b) r-v'-? =0; (c) : + Iv? = 4+1 (d) 30- T4=0
1.6 CYLINDERS AND QUADRIC SURFACES 9.Find the traces of the following surfaces in the planes parallel t0 the coordinate planes and identity each the surface. (a) ? v - ?=0; (b) r-v'-? =0; (c) : + Iv? = 4+1 (d) 30- T4=0...
5 answers
Uestion 5 remaining: Marked out of L0OFlag questionEnter the abbreviated electron configuration for a neutral 180 atomAnswer:Check
uestion 5 remaining: Marked out of L0O Flag question Enter the abbreviated electron configuration for a neutral 180 atom Answer: Check...
5 answers
6) Draw a detailed curved-arrow mechanism for sulfonation of benzene by electrophilic aromatic substitution, and include the given steps (12 points) Generation of the electrophile from sulfuric acid b. Nucleophilic attack on the electrophile All resonance structures of the cationic intermediate d. Restoration of aromaticity and product formationSOaHzSO4
6) Draw a detailed curved-arrow mechanism for sulfonation of benzene by electrophilic aromatic substitution, and include the given steps (12 points) Generation of the electrophile from sulfuric acid b. Nucleophilic attack on the electrophile All resonance structures of the cationic intermediate d. R...
5 answers
The IUPAC name ofthe following - cumpound33-dimethyl-S-hexne ~5-dimethyl-I-hevne E3-dimethyl-Z-hexyne 5,6-dimethyl-2-hcptyne product the following reaction? 1S. What is the major NaNH_NH,CH;CH,BrCH]CH;CHzCH;CH}
The IUPAC name ofthe following - cumpound 33-dimethyl-S-hexne ~5-dimethyl-I-hevne E3-dimethyl-Z-hexyne 5,6-dimethyl-2-hcptyne product the following reaction? 1S. What is the major NaNH_NH, CH;CH,Br CH] CH; CHz CH; CH}...
5 answers
QueSTION 36 Two numeric variables Xand Y exhibit _ A the 8 values . rezaune 0f Xand F (oTebme Yalways have E B the the Ste value 0terere ofyE 'tends t0 increase . 8s the Cthe WueotXinjess value of Y tends = to decrease Js the va ue 'otxaess D.the value of Ytends = t0 increjse Js the vabe otXdxreke
QueSTION 36 Two numeric variables Xand Y exhibit _ A the 8 values . rezaune 0f Xand F (oTebme Yalways have E B the the Ste value 0terere ofyE 'tends t0 increase . 8s the Cthe WueotXinjess value of Y tends = to decrease Js the va ue 'otxaess D.the value of Ytends = t0 increjse Js the ...
5 answers
Let $m$ and $n$ be two positive integers greater than 1 . If $lim _{alpha ightarrow 0}left(frac{e^{cos left(alpha^{n}ight)}-e}{alpha^{m}}ight)=-left(frac{e}{2}ight)$then the value of $frac{m}{n}$ is
Let $m$ and $n$ be two positive integers greater than 1 . If $lim _{alpha ightarrow 0}left(frac{e^{cos left(alpha^{n} ight)}-e}{alpha^{m}} ight)=-left(frac{e}{2} ight)$ then the value of $frac{m}{n}$ is...
5 answers
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.$$ f(x)-9^{operatorname{lan} x}$$
Determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section. $$ f(x)-9^{operatorname{lan} x} $$...
5 answers
Draw the structure of the compound CyH,O that might exhibit the "C-NMR spectrum below: Impunity Paaks Jc omitted from the Rk Lit Thc triplet at 77 ppm i$ CDCI;onetenencDEA4oiaChemical ahdl (poin)You do not have consider stercochemistn You do not have expliciily draw atom: Do not include Ione pairs in your anSWcE: They will nt be considered in thc grading:JaucunINoxt)
Draw the structure of the compound CyH,O that might exhibit the "C-NMR spectrum below: Impunity Paaks Jc omitted from the Rk Lit Thc triplet at 77 ppm i$ CDCI; onetenen cDEA 4oia Chemical ahdl (poin) You do not have consider stercochemistn You do not have expliciily draw atom: Do not include Io...
5 answers
Let Dbe the solid region enclosed by te spberex? +y? + (z + 3)2 =9.Our aim is t0 use triple integral in spherical coordinates to find the volume ofD. By taking the order of integration as dpdOde,the limits of integration of @ areOsosnW2sosn0s0sn/2None of these
Let Dbe the solid region enclosed by te spberex? +y? + (z + 3)2 =9.Our aim is t0 use triple integral in spherical coordinates to find the volume ofD. By taking the order of integration as dpdOde,the limits of integration of @ are Ososn W2sosn 0s0sn/2 None of these...
5 answers
Let X be a random variable with Uniform (0,2) distribution. Whatis the pdf of 𝑌=1/x? Use two different methods and compare
Let X be a random variable with Uniform (0,2) distribution. What is the pdf of 𝑌=1/x? Use two different methods and compare...
5 answers
TUn} Kd lubt Hobt H 1 8 ji' D 2 0J 31 1 { 3 3 1 8 I 3 1 3 [ 1
TUn} Kd lubt Hobt H 1 8 ji' D 2 0J 3 1 1 { 3 3 1 8 I 3 1 3 [ 1...
5 answers
VxIt is believed that about 70% of students prefer fast food tohealthy food. Of a random sample of 192 university students,123 said that they preferred fast food. Is there evidencethat it is not 70% ? Use a level of significance of5%.A. The p-value is 0.0098; Yes. B. The p-value is 0.0049; Yes. C. The p-value is 0.0351; Yes. D. The p-value is 0.0351; No. E. None are correct.
vx It is believed that about 70% of students prefer fast food to healthy food. Of a random sample of 192 university students, 123 said that they preferred fast food. Is there evidence that it is not 70% ? Use a level of significance of 5%. A. The p-value is 0.0098; Yes. B. The p-value is 0.0049...
5 answers
Ecomuversing camptonAelmonFucaald M-CFTeJAccol?Hoanin KFaenumbc nloces @hcunJ OenAcon pmphi0 Conyani02,360 ttFynahDtTaM ( Wttaluor-ug? [oo: Golo
Ecom uversing campton Aelmon Fucaald M-CFTeJ Accol? Hoanin KFae numbc nloces @hc unJ Oen Acon pmphi0 Conyani 02,360 tt Fynah DtTaM ( Wttalu or-ug? [oo: Golo...
5 answers
FCSuncscCEZANAOdErcokiCOS AAonp YNNeTMaTlenJesKinwind Tunsdri=WaictmWeocedttIctEancdtoennsTn '[C ICottyCY 12et ieecv0k Gad 105 5apo htay Tta Tn OcreJotn 678 0ql=CSE EC+Eotecein mnicnz Erocgin 0O77o? EnG Daied D7e Tejrg Viudsuemneconc II :J5 3o4adu E _ub ziilninoeoti f2t 06{3- 30cico [3724o2Ea 4nazn &lcereditaGrtto Rci Maguieoetncmtanahlouead CTcnraot IotCCTec EEEctd
FCSuncsc CEZANAOd Ercoki COS AAonp YNNe TMaTlen Jes Kinwind Tunsdri= Waictm WeocedttIct Eancdtoenn sTn '[C ICottyCY 12et ieecv0k Gad 105 5apo htay Tta Tn Ocre Jotn 678 0ql= CSE EC+E otecein mnicnz Erocgin 0O77o? EnG Daied D7e Tejrg Viudsuemneconc II :J5 3o4adu E _ub ziilninoeoti f2t 06{3- 30ci...
5 answers
Marginal cost for a product is -30 when producing 600 items and is 0 when producing 900 items If marginal cost is always increasing; then the total cost function has.relative minimum at 90Q items__a relative maximum at 900items.-relative minimum at 600 items_~orelative extrema_relative maximum at 600items.
Marginal cost for a product is -30 when producing 600 items and is 0 when producing 900 items If marginal cost is always increasing; then the total cost function has. relative minimum at 90Q items_ _a relative maximum at 900items. -relative minimum at 600 items_ ~orelative extrema_ relative maximum ...
5 answers
WHW_0I pdfWHW_02pdfWHW_03.pdffile ///C /Users/Owner/Desktop WHW_O1pdfFit to pagePage viewRead aloudAdd notesBusiness Calculus (MATH 'Z350) Written HomeworkPage 2 of 8Due Thursday; 06/ 202010 points| Find limexplain why thc limnit docs not exist:(2+2 il € < 0 10 points | Mf f(z) = find if z > 0" Iivg f (2). 2 - rType hiere t0 search
WHW_0I pdf WHW_02pdf WHW_03.pdf file ///C /Users/Owner/Desktop WHW_O1pdf Fit to page Page view Read aloud Add notes Business Calculus (MATH 'Z350) Written Homework Page 2 of 8 Due Thursday; 06/ 2020 10 points| Find lim explain why thc limnit docs not exist: (2+2 il € < 0 10 points | M...

-- 0.018831--