You are given that T:U→X is a linear transformation. Prove that, T(c;u, + C2U2 + · = c, T(u1) + c2T(u2) + · · · + C„T(Un) for all u; e U and for all c, e Rand n2 1. ...+ Cnun)

Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.

Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.

Show that T from Exercise 71 is represented by the matrix A=[12121212]. Proof Let T be the function that maps R2 into R2 such that T(u)=projvu, where v=(1,1) (a) Find T(x,y) (b) Find T(5,0) (c) Prove that T is a linear transformation from R2 into R2.

Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.

Let T:R3R3 be the linear transformation that projects u onto v=(2,1,1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of T.

Let T:P2P3 be the linear transformation T(p)=xp. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3}.

all of part (a) please

Calculate 121x dxdy where R is the parallelogram with vertices (0,0), (6,2), (7,6) R 1 -(v — Зи), у 11 -(4v – u) 11 and (1,4) using the transformation x =

Let D, be the linear transformation from C'[a, b] into C[a, b] from this example. Determine whether the statement is true or false. Explain. Dfext + 2x) = D,(e**) + 20,64) O False, D, is a linear transformation but does not preserve or produce non-linear transformations. O True, D, is a linear transformation and preserves scalar addition and multiplication. O False, D, is a linear transformation and preserves scalar addition and multiplication. O True, D, is a linear transformation and preserves addition and scalar multiplication.