Find a linear transformation T from R4 to R4 such that the kernel of T has dimension 2 and give a basis for the range of T. What is a basis for the kernel of T?

Let Let f: R² R2 be the linear transformation defined by [f]g = f(x) = {(-1,2), (2,-5)}, {(-1,2), (-1,3)}, be two different bases for R 2. Find the matrix [f] for f relative to the basis in the domain and in the codomain. = -5 5 -5 =

Find a basis for the kernel and the image of the linear transformation T: R³ R³ given by T(x) = Ax, where 1 2 3 A= -1 -2 -3 13 4

Show that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.

defined by Let Let ƒ : R² → R² be the linear transformation [f] B = B C f(x) = = = 3 2 -2 - -3 x. be two different bases for R². Find the matrix [f] for f relative to the basis B in the domain and C in the codomain. {(-1,2), (3,-7)}, {{-1, -2), (3, 7)},