Let Find a basis for the kernel of A (or, equivalently, for the linear transformation T(z) = Ar). A basis for the kernel of A is A = 4 -12 12 -4 -3 12 9 -12 -9

Consider the following linear transformation T: R5 → R3 where T(x1, X2, X3, X4, Xs) = (X1-X3+X4, 2x1+X2-X3+2x4, -2x1+3x3-3x4+Xs) (a) Determine the standard matrix representation A of T(x). (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of T(x). (d) Is T(x) one-to-one? Is T(x) onto? Explain. (e) Is T(x) invertible? Explain

A A basis for the image of A is - 1 2 8 3 -2 O 10 3 -2 0 - Find a basis for the image of A (or, equivalently, for the linear transformation T(x) = Ax).

1. Consider a linear transformation R² → R² described by a matrix 11 18 -6 -10| and a non-standard basis B of R²: A = A. Present a vector w 4-3 4-N = 3 in the coordinates of B, that is, find [w]. B. Find a matrix B describing the transformation A relative to the basis B. C. Use this result in order to compute A³ (don't use a calculator!). Reminder: [J-1] [a b] = c d ad - bc d a