4. (a) Show that {(2, 0, 1, 1), (-1,2,3, 1), (1,0,2, -1)} is a linearly independent subset of R¹. (b) Show that 0 1 3 10-2 2-3 0 {( 1 ) · ( 2 )·()} {(° 130 is a linearly independent subset of R2×3 (c) Show that {e-1, 2-4, sin(xx)} 1 is a linearly independent subset of the space V of functions from R to R.
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- Find those constants a, b, c, d such that f(z) = ax² + bxy + y? + i(x² + cxy + dy²)13.) Which of the following sets contain functions that are linearly independent? * А.) (3х — 1, х + 2, 4x) B.) (4, 2x – 3) O C.) [3 e^(2x),4 e^(2x)] О D) [2 e (3х - 1), - 3 е^(3х — 1)14) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
- Find curl F if F(x,y,z) = 16y°z5 i - 8x³z8j - 7xy4 k. curl F = (-28xy3 + 64x³z7) i + (7y4 + 80y°z4) j + (-24x²z8 - 80y4z5) k 3,7 O curl F = (-28xy + 64x°z7)i + (7y4 + 80y5z4) j + (24ײz8 + 80y4z5) k O curl F= (-28xy3 + 64x³z7) i+ (7yª + 80y5,4) j + (24x²z8 - 80y4z5) k = = (28xy³ + 64x³z7) i+ (7y^ + 80y5z4) j + (-24x²z8 - 80y4z5) kWhich of the following pair of functions is linearly independent? (а) еЗx, е(х-3) (b) 4, sin?x + cos?x (c) 2sin2x, sinxcosx (d) sinhx, (e* – e*)3) Consider stress of a body at a particular point is give in the form of Q(x1, x2, X3) = xỉ - x – 4x1x2 + 4x2x3. Find an orthogonal change of variable that eliminates the cross product terms in Q(x1, X2, X3). (11)
- Find curl F if F(x,y,z) = 16y5₂5i - 8x³₂8 j- 7xy4k. curl F = (-28xy³ + 64x³₂7) i + (7y4 + 80y5z4)j + (-24x²z8 - 80y4₂5) k O curl F = (-28xy³ + 64x³z7)i + (7y4 + 80y5z4)j + (24x²z8 + 80y4₂5) k curl F = (28xy3 +64x³₂7) i + (7y4 + 80y5z4) j + (-24x²z8-80y4₂5) k O curl F = (-28xy3 + 64x³₂7)i + (7y4 + 80y5₂4)j + (24x²z8-80y4z5) kUse the general chain rule to find DH(1,-1) given √u √u² +3v² √v+2 G(U)=G(u, v) = and H(U)= F(G(U)). + + 2z² F(X) = F(x, y, z) = [ ²² 25,7² ›=[*******].Show that x and y are orthogonal in R^n if and only if x + y = x − y
- Is the set of functions f1(x) = ex+2, f2(x) = ex-3 linearly dependent or linearly independent on (−∞, ∞)? Explain.Consider the simple linear model Y, = Be + Bix, + C. where --N(0,a); i = 1,2, .., n. 1. Show that a) Y-IIN(B + B1*i,g²) b) (i) L = L(B.Pa:y) = (2ro) žexp - - Po + B,x,| (ü) a = (Sx - PSy) c) Cov(Be.B) = d) B, is an unhiased estimator of B, and that Var(B) = o 2. Derive the statisic for testing Ho: B - Bio versus H,: ß1 + Pin. (You must also state the distribution and critical region for the lest). 3. Constract a 100(1 – a)% confidence interval for 8,. %3D (Show all results used as part of your answer) %3Da) Prove the one point rule - 3-version: (3x)(x = t ^ A) = A[x := t] if x is not free in t. b) Prove the dual result 6 on p.176. Prove (3x) AvBC = (3x) ÂС V (3x) BC. (Hint: Translate first to Standard notation) c) Prove using the auxiliary variable metatheorem: + (3x) (A → B) → (Vx)A → (3x)B