S In واں U₁₂ U₂ are two TRY. U₁ = one U₂ = 3 2 3 (0000) 2 2 base surfaces 2 3 1000 3 Determine U₁₂ U₂, U₁ nu₂ > V₁ + V₂ each and dimension 2 Also determine a surface W of TR såh that U₁&W=R²" 4
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- Show that except in degen-erate cases, (u * v) * w lies in the plane of u and v, whereas u * (v * w) lies in the plane of v and w. What are the degenerate cases?Consider the following geometry problems in 3-spaceEnter T or F depending on whether the statement is true or false. 1. Two lines either intersect or are parallel 2. A plane and a line either intersect or are parallel 3. Two planes orthogonal to a third plane are parallel 4. Two lines orthogonal to a third line are parallel 5. Two planes parallel to a line are parallel 6. Two lines parallel to a third line are parallel 7. Two lines parallel to a plane are parallel 8. Two lines orthogonal to a plane are parallel 9. Two planes orthogonal to a line are parallel 10. Two planes either intersect or are parallel 11. Two planes parallel to a third plane are parallelUse greens theorem to evaluate line interal (x2y)dx+y2dy),where c is the closed path formed be y=x and y=x3 from(0,0) to (1,1)
- An _____ is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two fixed points called _____ is a constant.What vector space property does Q2 ={(m, n) m, n ∈ Q} fail? *set Q: rational numbersVerify the Cauchy-Schwarz Inequality for u = (1, −1, 3) and v = (2, 0, −1).
- Consider the empty set, i.e. X={ }. Which property of vector space is not satisfied and how?Setx = [ 0 : 4, 4,−4, 1, 1]' and y = ones(9, 1) Use the MATLAB function norm to computethe values of ||x||, ||y||, ||x + y|| and to verifythat the triangle inequality holds. UseMATLABalso to verify that the parallelogram law||x + y||2 +||x − y||2 = 2(||x||2 + ||y||2)State true or false with a brief justification If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional
- Show that the spaces (R,U) and (R,S) are Hausdorff.Evaluate the least distance of the plane below from the origin in Euclidean space : x−2y+2z=5x−2y+2z=5 Select one: 2/3 4/3 7/3 5/3You are now allowed to assume that the half-planes determined by the line with the equation ax+by +c = 0 correspond to the points (x, y) so that ax + by + c < 0 and ax + by + c > 0, respectively. Usingthis, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given lineand that (s, t) and (u, v) are on the same side of the given line. en construct the parametrized linethrough (q, r) and (u, v). Consider the mappingλ γ7→ a(q − qλ + uλ) + b(r − rλ + vλ) + c and note that it is continuous and either increasing or decreasing. Use this fact to show that, for everyλ, γ(λ) > 0 or γ(λ) < 0, depending on which half-plane the points are on.)