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Let V be a three-dimensional vector space with ordered basis B = {u, v, w}. Suppose...

Let V be a three-dimensional vector space with ordered basis B = {u, v, w}. Suppose that T is a linear transformation from V to itself and T(u) = u + v, T(v) = u, T(w) = v.

1. Find the matrix of T relative to the ordered basis B.

2. A typical element of V looks like au + bv + cw, where a, b and c are scalars. Find T(au + bv + cw). Now that you know what the image of an arbitrary element of V is, write down a basis for the kernel of T. (Your basis elements will of course be expressed in terms of u, v and w.)

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