Exercise 2. Prove that dim(null T") = dim(null T) + dim W – dim V and dim(range T") = dim(range T), for every T E L(V, W).

Q: Differentiate the function. f(x) = sin(x) In(4x) f'(x) =

A: ddxu(x)v(x)=v(x)ddx(u(x))+u(x)ddx(v(x)) ddx(sin(x))=cosx ddx(ln(ax))=aax=1x…

Q: Solve 2sinx- V3 = 0 on interval [0,2 t)

A: We have to solve  2sinx - √3 = 0 on interval [0 , 2π)  Since ,  2sinx - √3 = 0  2sinx = √3

Q: 6) Find R100 for y = x³ on the interval [0,2]. Use four decimals.

A:

Q: For which values of ß does {(1,0, ß),(1,2,–3)(B,1,0)} span R'?

A: If spans the same subset of R³, the vectors must be linearly dependent to each other.    Therefore,…

Q: Which of the following pair of functions is linearly independent? (a) e3*, e(x-3) (b) 4, sin?x +…

A:

Q: if r'y-- 2y + 1.

A:

Q: Calculate the derivative of the function. Then find the value of the derivative as specified. dr 6.…

A:

Q: If f.(x, y) = 13x + 8y + 11, || f.(3, 2) = a. b. f„(5, 2)

A: Given, fx(x,y) = 13x + 8y + 11 a). fx(3,2) = 13*3 + 8*2 + 11 = 39+16+11 => fx(3,2) = 66

Q: Find those constants a, b, c, d such that f(z) = ax² + bxy + y² + i(x² + cxy + dy?)

A: The given function is : fz=ax2+bxy+y2+i(x2+cxy+dy2)

Q: find Pn (Q'U R).

A:

Q: find range of y 1 1 X-2 R/{1,2} O R/{2} O R/{1} O R/{0,1} O 1-

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.