gibson:teaching:fall-2016:iam961:hw1

**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).

gibson/teaching/fall-2016/iam961/hw1.txt · Last modified: 2016/09/07 08:56 by gibson