Need help with these math problems, may someone show the work for it, please. Thank you.

Assume that T is a linear transformation. Find the standard matrix of T. T: R2 -*, T(e1 ) = (6, 1, 6, 1), and T (e2) = (- 8, 5, 0, 0), where e, = (1,0) and ez = (0, 1 ) . . . . A = (Type an integer or decimal for each matrix element.)Assume that T is a linear transformation. Find the standard matrix of T. T: R2 -R, rotates points (about the origin) through a radians. . . . A= (Type an integer or simplified fraction for each matrix element. Type exact answers, using radicals as needed.)Assume that T is a linear transformation. Find the standard matrix of T. T: IR32 be a linear transformation such that T09 x2) = (x1 +x2: 2x1 + 3x2) . Find it such that TM 2 (5:6). 0 12 C Let A = 0 0 U= 18 and v = b . Define T: R* -R* by T(x) = Ax. Find T(u) and T(V). -24 d 0 0 T(u) = (Simplify your answer. Use integers or fractions for any numbers in the expression.)- 3 2 If T is defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A = - 4 0 1 - 2 and b = -3 2 4 Find a single vector x whose image under T is b. X =]1 - 4 - 4 If T is defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A = - 3 4 - 4 and bang Find a single vector x whose image under T is b. X =Let A be a 7 x 4 matrix. What must a and b be in order to define T : R-R by T(x) = Ax? . . . a = (Simplify your answer.)Find all x in R* that are mapped into the zero vector by the transformation XIAx for the given matrix A. -4 12 -4 A= 0 1 -5 3 4 - 12 28 - 4 Select the correct choice below and fill in the answer box(es) to complete your choice. O A. There is only one vector, which is x = OB. X3 + X4 OC. X1 + X2 + X4 OD. X3Let T: R2 ->IR be a linear transformation that maps u = Use the fact that T is linear to find the images under T of 4u, 2v, and 4u + 2v. The image of 4u is4 X1 Lete, = andez = 1 y , = ,andy2 = , and let T: 12 - 12 be a linear transformation that maps e, into y, and maps e, into yz. Find the Images Of and X 2 The image of isShow that the transformation T defined by T(X,, X2) = (3X, - 2X2, X, + 3, 5X2) is not linear. If T is a linear transformation, then T(0) = and T(cu + dv) = cT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d. (Type a column vector.)