Ler L: R4 → R3 be the linear transformation defined by (4p) L(z,y,z, t) = (x – y +t, 2x – 2, Y + 2z – t) a) Find the images of the standard basis of RA L(1,0,0,0) = L(0,1,0,0) = L(0,0,1,0) = L(0,0,0,1) = b) Find a basis and the dimension of the image of L c) Find a basis and the dimension of the kernel of L (8p) (8p)
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
1. Consider the linear map f R3R4 defined by f(x, y,z) (x+y+ z, 2x +4z,3x + 2y +4z, 5y - 5z) a.) Find the matrix representing f (5pts) b.) Determine (i) ker(f) (2pts) (ii) Range(f) (2pts) and (i) dim(f) (lpt)
Consider the vector field F(x, y, z) = 8x^2 + 3y, −5x^2y − 4y^2, 6x^2 + 7y − 8 which is defined on all of double-struck R3, and let F be the rectangular solid region F = {(x, y, z) | 0 ≤ x ≤ a, 0 ≤ y ≤ b, −1 ≤ z ≤ 1} where a > 0 and b > 0 are constants. Determine the values of a and b that will make the flux of F...
(2x + y - z = -12 3 -x + y + z = 7 1-2x + 4z = 34
1. Let L: R2-R2 be defined by L(x.y) (x +2y, 2x - y). Let S be the natural basis of R2 and let T = {(-1,2), (2,0)) be another basis for R2 . Find the matrix representing L with respect to a) S b) S and1T c) T and S d) T e) Find the transition matrix Ps- from T basis to S basis. f) Find the transition matrix Qre-s from S-basis to T-basis. g) Verify Q is inverse of...
Suppose Cor(X,Y)=1/3 and oy = 403. Let Z=2X +3Y. If Var(Z) = 240, calculate Var(X).
12х + 18у — 4z. (1 point) Let x, y, z be (non-zero) vectors and suppose w = If z 2x 3y, then W = X+ у. Using the calculation above, mark the statements below that must be true. |A. Span(w, x) Span(w, z) B. Span(w, y) Span(w, y, z) |C. Spanx, y, z) = Span(x, y) D. Span(w, z) Span(x, y) E. Span(w, x, z) = Span(w, x, y)
a. Sketch the solid S:= {[x; y; z] in |R3 | x,y,z ≥ 0, and 2x + 4 y + 2z ≤ 12}. b. Using your calculator evaluate, i) as a triple integral and ii) by the divergence theorem, the volume of S. c. Find i)the surface area of the solid S and ii)the flux thru the top of S due to the vector field F, where F(x,y,z) = ( x + yz , y + xz , z +...
Let V be the set of vectors [2x − 3y, x + 2y, −y, 4x] with x, y R2. Addition and scalar multiplication are defined in the same way as on vectors. Prove that V is a vector space. Also, point out a basis of it.