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in a vector space v over R, prove that (-1)X = -7 Hind (you may use...
6. (i) Prove that if V is a vector space over a field F and E is a subfield of F then V is a vector space over E with the scalar multiplication on V restricted to scalars from E. (ii) Denote by N, the set of all positive integers, i.e., N= {1, 2, 3, ...}. Prove that span of vectors N in the vector space S over the field R from problem 4, which we denote by spanr N,...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
4. Prove that a vector space V over F is isomorphic to the vector space L(F,V) of all linear maps from F to V. Note: We are not assuming V is finite-dimensional.
Suppose that V is a 3-dimensional vector space over a field F and T : V → V is a linear tion such that the corresponding F[x]-module structure on V is given by 7. V F[x]/(x3-x2-x + 1). Among the matrices A, B, and C given below, which are the matrix of T in some basis for V. Explain 1 1 0 0 0-1 B-10 1 A 0 1 0 0 1 1 0 0 -1 0 0 -1 (Note:...
Let V be the vector space consisting of all functions f: R + R satisfying f(x) = a exp(x) +b cos(x) + csin(x) for some real numbers a, b, and c. (The function exp refers to the exponential, exp(22) = e.) Let F be the basis (exp cos sin of V. Let T :V + V be the linear transformation T(f) = f + f' + 2f" (where f' is the derivative of f). You may use the linearity of...
Problem 4. Let n E N. We consider the vector space R” (a) Prove that for all X, Y CR”, if X IY then Span(X) 1 Span(Y). (b) Let X and Y be linearly independent subsets of R”. Prove that if X IY, then X UY is linearly independent. (C) Prove that every maximally pairwise orthogonal set of vectors in R” has n + 1 elements. Definition: Let V be a vector space and let U and W be subspaces...
(7) Let V be a finite-dimensional vector space over F, and PE C(V) In this question, we will show that P is an orthogonal projection if and only if P2P and PP It may be helpful to recal that P is the orthogonal projection onto a subspace U if and only if (1) P is a projection, and (2) ran(P)-U and null(P)U (a) Prove that if P is an orthogonal projection, then P2P and P is self-adjoint Hint: To show...
A weird vector space. Consider the set R+ = {x ER : x > 0} = V. We define addition by x y = xy, the product of x and y. We use the field F = R, and define multiplication by co x = xº. Prove that (V, O, RO) is a vector space.
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
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