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# 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 #### 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? 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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...
##### 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...
##### Let X be a random variable with Uniform (0,2) distribution. Whatis the pdf of ð‘Œ=1/x? Use two different methods and compare
Let X be a random variable with Uniform (0,2) distribution. What is the pdf of ð‘Œ=1/x? Use two different methods and compare...
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TUn} Kd lubt Hobt H 1 8 ji' D 2 0J 3 1 1 { 3 3 1 8 I 3 1 3 [ 1...
##### 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...
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