Let A = A basis for the image of A is -1 −1 1 1 -6 -3 3 6 2 2 -2 -2 Find a basis for the image of A (or, equivalently, for the linear transformation T(x) = Ax).

(4) Define a linear transformation T : R³ → R³ by 8-8-8-8-8-8 = = Is this transformation invertible? What does this transformation do to a vector U ? (Bonus: if you write a polynomial as a + bx + cx², how does that relate to the transformation?)

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

Let B = {b₁,b2,b3} be a basis for vector space V. Let T: V → V be a linear transformation with the following properties. == T(b₁) = -6b₁ + 5b₂, T(b₂) -3b₁ +4b₂, T (b3) =b₁ 1 Find [T]B, the matrix for T relative to B.

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).

Consider the linear transformation T: C→C of the last problem. Find the matrix T that represents it in each of the following basis: a) {1, i}, b) {1+i, 1+2i}.

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