Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn. Let pEN be a prime and let V be a vector space over Zp with dimension n. Show that |v| = pn.
Let m, n EN\{1}, V be a vector space over R of dimension n and (v1, ..., Vm) be an m tuple of V. (Select ALL that are TRUE) If m > n then (v1, ..., Vm) spans V. If (v1, ..., Um) is linearly independent then m <n. (v1, ..., Um) is linearly dependent if and only if for all i = 1,..., m we have that U; Espan(vi, .., Vi-1, Vj+1, ..., Um). Assume there exists exactly one...
9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such that f-f. a) Show that fis diagonalizable and that V-rge f b) Determine x and . f ker f. 9.4 (Idempotent endomorphisms) Let V be an n-dimensional vector space over Kand let End(V) such that f-f. a) Show that fis diagonalizable and that V-rge f b) Determine x and . f ker f.
10. Let [n] be an element in Zp, p prime. We say [n] is perfect provided [o (n)] [2n]. Show that d-[21,where Idy]-'is the multiplicative inverse of ld in Z, [dkl In 10. Let [n] be an element in Zp, p prime. We say [n] is perfect provided [o (n)] [2n]. Show that d-[21,where Idy]-'is the multiplicative inverse of ld in Z, [dkl In
Let V be a vector space over R and let v1, ..., Un each be a vector in V \{0}. Show that (v1, ..., Un) is linear independent if and only if span(V1, ..., vi) n span(Vi+1, ..., Un) = {0} for all i = 1,...,n - 1
Q4 6 Points Let V be a vector space over R and let V1, ... , Vn each be a vector in V \{0}. Show that (v1, ..., Vn) is linear independent if and only if span(v1, ... , Vi) n span(vi+1, ..., Vn) = {0} for all i = 1,...,n - 1
4. Let v={[a -.:a,nccc} Note that V is a vector space over R. View V as a R-vector space. (a) Find a basis for V over R. (b) Let W be the set of all matrices M in V such that M21 = -M12, where denotes complex conjugate. Show that W is a subspace of V over R and find a basis for Wover
Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3. Problem 4. Let p be an odd prime, and let Tp C Zp denote the set of elements of Zp which are perfect cubes: Tp-(a: a E z;} (1) Show that if p1 (mod 3) then Tp (p 1)/3.
(1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1)) (b) Let φο : Zp[2] Zp be the evaluation homomorphism at 0. Compute φο(zp-1-1) and φο((1-1)(1-2) . . . (z-(p-1))) (c) Use parts (a) and (b) to conclude that (-1)--1. (1) Let p be a prime number. The following polynomials are considered as elements in Zp[ (a) Show that zP-1-(z -1)( 2) ( (p 1))...
Let W be a subspace of an n-dimensional vector space V over C, and let T:V V be a linear transformation. Prove that W is invariant under T if and only if W is invariant under T- I for any i EC.
Q4 6 Points Let V be a vector space over R and let Vi, ..., Ur each be a vector in V\{0}. Show that (v1,..., Vre) is linear independent if and only if span(v1,..., vi) n span(Vi+1,...,Vn) = {0} for all i = 1,...,n-1 Please select file(s) Select file(s)