1. Prove that any linear map can written as an matrix. (Hint: let . Express as a linear combination of the canonical basis vectors . Substitute that into , then use linearity to rewrite the right-hand-side of this equation as a linear combination of vectors . Now take the inner product of both sides of this equation with . That should give you for some matrix .)
2. Prove that .
3. If and are -vectors the matrix is known as a rank-one perturbation of the indentity. Show that if is nonsingular, then its inverse has the form for some scalar , and give an expression for . For what and is singular? If it is singular, what is ? (Trefethen exercise 2.6).