Let V be an inner product space For a fixed nonzero vector v

Let V be an inner product space. For a fixed nonzero vector v_0 in V, let T. V rightarrow R be the linear transformation T(v) = (v, v_0). Find the kernel, range, rank, and nullity of T. Calculus Let B = {1, x, sinx, cos x} be a basis for a subspace W of the space of continuous functions, and let D_x be the differential operator on W. Find the matrix for D_x relative to the basis B. Find the range and kernel of D_x. Writing Under what conditions are the spaces M_m, n and M_p, q isomorphic? Describe an isomorphism T in this case. Calculus Let T: P_3 rightarrowP_3 be represented by T(p) = p(x) + p\'(x). Find the rank and nullity of T. The Geometry of Linear Transformations in the Plane In Exercises 73-78, identify the transformation and graphically represent the transformation for an arbitrary vector in the plane. T(x, y)=(x, 2y) T(x, y)=(x + y, y) T(x, y)=(x, y + 3x) T(x, y)=(5x, y) T(x, y)=(x + 2y, y) T(x, y)=(x, x + 2y) In Exercises 79-82, sketch the image of the triangle with vertices (0, 0), (1, 0), and (0, 1) under the given transformation. T is a reflection in the x-axis. T is the expansion represented by T(x, y) = (2x, y). T is the shear represented by T(x, y) = (x + 3y, y). T is the shear represented by T(x, y) = (x, y + 2x). In Exercises 83 and 84, give a geometric description of the linear transformation defined by the matrix product. [0 2 1 0] = [2 0 0 1] [0 1 1 0] [1 0 6 2] = [1 0 0 2] [1 0 3 1] Computer Graphics In Exercises 85-88, find the matrix that will produce the indicated rotation and then find the image of the vector (1, - 1, 1). 45 degree about the z-axis 90 degree about the x-axis 60 degree about the x-axis 30 degree about the y-axis In Exercises 89-92, determine the matrix that will produce the indicated pair of rotations. 60 degree about the x-axis followed by 30 degree about the z-axis

Solution

70.Let B={1,x,sinx,cosx} and w be a differential operator

Differentiate elements of B w.rt. x

d1/dx=0=1.0+x.0+0.sinx+0.cosx

dx/dx=1=1.1+0.x+0.sinx+0.cosx

d sinx/dx=cosx=1.0+x.0+0.sinx+1.cosx

d cosx/dx=-sinx=1.0+x.0+(-1)sinx+0.cosx

now take the coefficient of 1,x,sinx,cosx and arranged in a colomn we get required matrix.

The transpose of following matrix

[0,0,0,0;1,0,0,0;0,0,0,1;0,0,-1,0]

Reduced matrix into echelon form then column contains pivot elemnts is range spanned by vectors

Consider above augmented matrix [ Dx 0] and reduce into echelon form last we get system of linear equation

solution of system is basis for Kernel