VA-) is ,= (1, 1, 1), Wa= (1, 2, 3), W =(4,5,8) show that vectors stretch...
Question 1: Let T: R3 ---> R2 defined by T(x1,x2,x3) = (x1 + 2x2, 2x1 - x2). Show that T as defined above is a Liner Transformation. Question 2: Determine whether the given set of vectors is a basis for S = {(1,2,1) , (3,-1,2),(1,1,-1)} R3 Need answers to both questions.
Can I get help with questions 2,3,4,6? be the (2) Determine if the following sequences of vectors vi, V2, V3 are linearly de- pendent or linearly independent (a) ces of V 0 0 V1= V2 = V3 = w. It (b) contains @0 (S) V1= Vo= Va (c) inations (CE) n m. -2 VI = V2= V3 (3) Consider the vectors 6) () Vo = V3 = in R2. Compute scalars ,2, E3 not all 0 such that I1V1+2V2 +r3V3...
6. Given the vectors vi = - 0 -- --(2.).-) no estaba 1. vz = 2 .03 = 1 -1 1 62-5) ,0 = 3, find the value(s) of k so that: de (a) vis in Span{vi, v2, U3}. (b){i, 03, 03} form a linearly independent set. (c){vi, už, va} form a basis for R3. (d) span{ti, uz, va} is a plane in R.
?24) 1. To show that set of vectors of formm 2. (a) if A is invertible, list three different methods to solve equation Ax-b. b) Application each of above mentioned methods to solve is a subspace of a space of all 2x2 matrices. -x2 +2x3 =0 Find a matrik that reflects vectors in R' about yz-plane and then expand the length twice (2 0 3 2 7 Given set of vectors S=(1 1, 1-1, 1 .. D in R 10)...
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...
(7) Consider the set W of vectors of the form | 4a + 36 1 0 a+b+c c-2a where a,b,c E R are arbitrary real numbers. Either describe W as the span of a set of vectors and compute dim W, or show that W is not a linear subspace of R. (8) Find a basis for the span of the vectors 16115 1-1/ 121, ܘ ܟ ܢܝ
5 = 10 marks ] Question 1 [3 2 (a) Use the Fourier transform, -) / Ф(Р) e'pr/h d3p 27TH and the inverse transform 1 b(FeipF/hd3r Ф(Р) 2тh to prove the Fourier Integral Theorem: 1 ') ei(F'-p)/h d3p' d*r. Ф(р) - 2тh (b) Explain why the Dirac-ô may be represented via - ih)/ 1 8(F- F') (c) Show that for arbitrary wave functions /a,b(f) that / -/ Фа (р)" фь (р) d'р, Va(r = where ba and da (and /,...
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...
Exercise 1. Let v = 2 ER3. Recall that the transposed vector u is ū written in row form, 3 that is, of = [1 2 3]. It can be seen as a 1 x 3 matrix. For every vector R3, set f(w) = 1 WER. (i) Show that f: R3 → R defines a linear transformation. (ii) Show that f(ū) > 0. (iii) What are the vectors we R3 such that f(w) = 0?
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...