Exercise 1. Let v = 2 ER3. Recall that the transposed vector u is ū written...
1 1 0 -1 Exercise 2. Let A = 0 1 0 in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R”, set g(ū) = WT AV E R. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]c,b in the bases C = {1} and B = { 9 8 B |}? (ii) Let f: R3 + R be the function defined by f(w) = vſ Aw...
o 1 0 -1 Exercise 2. Let A= in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R3, set g(W) = WT AT ER. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]C,B in the - 1 bases C = {1} and B { 8.00 } ? (ii) Let f : R3 → R be the function defined by f() = 7T Aw E R. Show that...
(i) Find an orthonormal basis {~u1, ~u2} for S (ii) Consider the function f : R3 -> R3 that to each vector ~v assigns the vector of S given by f(~v) = <~u1, ~v>~u1 + <~u2; ~v>~u2. Show that it is a linear function. (iii) What is the matrix of f in the standard basis of R3? (iv) What are the null space and the column space of the matrix that you computed in the previous point? Exercise 1. In...
Vectors pure and applied Exercise 11.5.9 Let U and V be finite dimensional spaces over F and let θ : U linear map. v be a (i) Show that o is injective if and only if, given any finite dimensional vector space W map : V W such that over F and given any linear map α : U-+ W, there is a linear (ii) Show that θ is surjective if and only if, given any finite dimensional vector space...
Hi, could you post solutions to the following questions. Thanks. 2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that...
I need the answer to problem 6 Clear and step by step please Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a linear transformation. 2. Show that aT is a linear transformation for any scalar a. 3. Suppose that T is invertible. Show that T-1 is also a linear transformation. Problem 5. Let T : R3 →...
Please answer me fully with the details. Thanks! Let V and W be vector spaces, let B = (j,...,Tn) be a basis of V, and let C = (Wj,..., Wn) be any list of vectors in W. (1) Prove that there is a unique linear transformation T : V -> W such that T(V;) i E 1,... ,n} (2) Prove that if C is a basis of W, then the linear transformation T : V -> W from part (a)...
1. Let T be the matrix T=10 3 acting on the complex vector space V C3 (a) Recall how T defines the structure of a C-module on C3. (b) Let p(x71, and let 2Compute the element p(x) v of C3 (c) Give a set of generators and relations for C3 over Cz] with the above module structure. (d) Write down the relations matrix (e) Recall the definition of minimal polynomial of a matrix. (f) What is the minimal polynomial of...
2 5 Do the vectors u = and v= 3 7 span R3? -1 1 Explain! Hint: Use Let a, a2,ap be vectors in R", let A = [a1a2..ap The following statements are equivalent. 1. ai,a2,..,a, span R" = # of rows of A. 2. A has a pivot position in every row, that is, rank(A) Select one: Oa. No since rank([uv]) < 2 3=# of rows of the matrix [uv b.Yes since rank([uv]) =2 = # of columns of...
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of R'. Definition: (Modified from our book from page 204.) Let V be a subspace of R". Then the set of vectors (61, 72, ..., 5x} is a basis for V if the following two conditions hold. (a) span{61, 62,...,x} = V (b) {61, 62, ..., 5x} is linearly independent. Definition: Standard Basis for R" The the set of vectors {ēi, 72, ..., en) is...