Problem 1. Σvex (f+(v) - ƒ¯(v)) = a) Show that, for any flow f in a network N and any set X CV, f+(X) — ƒ¯(X). (15 points). b) Give an example of a flow f in a network such that Σvex ƒ+(v) ‡ ƒ+(X), and Σvex ƒ¯(v) ‡ ƒ¯(X). (10 points).
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- Determine whether the following sets are linearly dependent or linearlyindependent. (a) {(; )( )} . -9} in Maxa(R) 4 (b) {(- ).(- 9G -)} in M2x2(R) 4 (c) {x 3 + 2x2, -x2 + 3x + 1, x3 – x2 + 2x – 1} in P3(R) (d) {x 3 – x, 2x2 + 4, -2x3 + 3x2 + 2x + 6} in P3(R) (e) {(1, -1, 2), (1, -2, 1), (1, 1, 4)} in R3 (f) {(1, –1, 2), (2, 0, 1), (–1, 2, –1)} in R3 {(÷ ) ( -) (; 0. 2 ).(- )} in Max2(R) (g) (m) {(-; ') (? ). -2)} in M2x2(R) () (x 4 - х3 + 5x2 - 8х + 6, -х4 + х3 - 5х2 + 5х - 3, x 4 + 3x2 – 3x + 5, 2x4 + 3x3 + 4x2 – x + 1, x3 – x + 2} in P4(R) ) (x4 - х3 + 5x2 - 8х + 6, -х4 + x3 - 5x2 + 5х - 3, х4 + 3x2 - 3х + 5, 2х4 + х3 + 4x2 + 8х} in P4(R)The cantilever beam shown in Figure 2.1 is subjected to linearly distributed loads. a) Draw all necessary free-body diagrams using the method of sections and find symbolic expressions for the internal shear force (V(x)) and bending moment (M(x)) due to the external forces as functions of x, where x is a horizontal coordinate along the length of the beam, measured from the most left support of the beam. Your expression(s) must allow to obtain internal reactions for 0Which of the following functions are linear? Justify your answer in a comprehensible manner. In the case of a linear function, determine the basis of the image and the core and also the associated dimensions in a comprehensible manner. (i) fi: R R, (a, b,c)"() (ii) fa : R - R. (:) ) (ii) f; : R → R', (:) ) (iv) fa : R → R', (a, b,c) (1,1, 1)"8. Decide which of the following sets are linearly independent in R". Justify your answer in each case. (a) X₁ = = {(1,0), (0, 1)} ℃R² (b) X₂ = {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0,0)} ℃ R² (d) X4 = {(1,0), (0, 1), (1, 1)} C R² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} CR³ (f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (−1, 0, 1)} ℃ R³ (g) X7 = {(0,0,0), (0, 1, 0), (1, 1, 1), (–1, 0, 1)} ℃ R³ (h) X8 = {(0, 1,0), (1, 1, 1)} C R³ (i) X9 = {(4, 3, 0, 0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} ℃ R4 (j) X10 = {(1, 2, 0, 0), (0, 2, 3, 0), (0, 0, −1, 1)} ℃ Rª 9. Decide, for the sets X; above, for which i, j = {1, 2, ..., 10}, span(X₂) = span(X;).Are the following sets linear independent: (a) {p(x)=3x+2x², q(x) = −2+x+2x², r(x) = x + x²} (b) {u1 (1,2,3), u₂ = (3, 2, 1), uz = (0, 4, 8)} = Which of the following sets span R³ (a) S = {(1,1,0), (0, 1, 1)} (b) S = {(1,0, 1), (0, 1, 2), (−1, −4, 2)} Check if the set S is basis for the given vector space V. (a) S = {1, x, x², 2x² + x − 2} and V = P₂. -1 (b) S = {S₁ = ( 2 − 2) こ - , S2 = (32) }, and V = M2,2.3. Write the mapping rule for this function a) y = - 3f[-4(x - 1)] + 2: ) ay + c) (x, y) → + d, b) f(x) = |x +1 - (x, y) – (; + d, ay + e)Let p(x) =-2x* + c,x' + c,x + c,x + c,x E R be a polynomium whose graph insersects the points (-2, 4), (- 1, 0), (1,-2), (3, 6). (a) Determine the coefficients of c,, c,, c1, Co-5) Let g(2) = u(2)+ivl2) be Ol complete function Ond hiC →C, h( 2) = e¯( v(a)+iu(2). Show that if h is a full function, g is constant function on(a) Let M R. Give the radius r and the center c of B(-2,5) n B(6, 7). (b) Let M = R. Using interval notations, give a simplified expression for the set L= [B(1,1) UB(5,1)] n B(3,2). (c) Let = 3(-1)" + and A = {,: ne N). Give the set W of accumulation points of A. (d) Let A₁-(-2n, 2n) and K = U An. Give a simplified expression for the interior Kº of K. NEN (e) Give the set T of isolated points of B(0, 1) U (3) U (2,5,7) (f) Let [P,Q] be a segment in R2 with midpoint H, and let (1,0) and (2, 1) be the components of the points P and H, respectively. Give the components (x, y) of the point Q. (g) Suppose that f: (M, d) →R satisfies d(f(x), f(y)) - vl. Let 21 e M and set n+1 = f(n), for n e N". Assume that M is complete and let a = lim zn. Give a simple formula satisfied by a.SEE MORE QUESTIONS
