Please help me! Which of the following sets w are subspaces of V? Justify your a

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Please help me!

Which of the following sets w are subspaces of V? Justify your answers. V = p2 and W = {p epsilon p2; p(0) = 1} V= F(R,R) and W = {f epsilon V; f(x) = acos(x) + bsin(x) + c where a, b, c epsilon R} V = C(R,R) and W = {f epsilon V; f(1) = 0} Determine whether the given functions are linearly independent or not, forall x epsilon R, f1(x)= x2 + 2, f2(x) = 1 ndash 2x, and f3(x) = (x + l)2 f1(x)= sin (pi x), and f3(x) = COS(2pi x) Suppose x, y, z epsilon Rn are linearly independent, prove the (x + y), (x + z), (y + z) are linearly independent. For Problems 2.6 and 2.7. calculate the uniform load (in addition to the beam weight) that will cause the sections to begin to crack if they are used for 28-ft simple spans. fc' = 4000 psi. fr = 7.5 , and reinforced concrete weight = 150 lb/ft3. Transformed-Area Method For Problems 2.8 to 2.14, assume the sections have cracked and use the transformed-area method to compute their flexural stresses for the loads or moments given. Repeat problem 2.8 if four #6 bars are used, (Ans. fc = 1356 psi, fs = 26.494 psi) (Ans. fc = 1258 psi, fs = 14,037 psi in bottom layer. fs = 12,889 psi at steel centroid)

Explanation / Answer

7. (a) Not a subspace as 0 doesn't belong to W

(b) Not a subspace g,h belongs to W doesn't necessariliy imply g+h belongs to W

(c) Its a subspace

9.(a) They are linearly ind because

a*f1(x)+b*f2(x)+c*f3(x) = 0

=> a=b=c=0

(b)They are linearly ind because

a*f1(x)+b*f2(x)+c*f3(x) = 0

=> a=b=c=0

10. a*(x+y) + b*(x+z) + c*(y+z) = 0

=> x*(a+b)+y*(a+c)+z*(b+c)=0

=> a+b=b+c=c+a=0

as x,y,z are lin ind

=> a=b=c=0

Hence x+y,y+z,z+x are linearly independent

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