18.7. EXERCISES 459

(e) Show there is at most one such w.

You have now proved the Riesz representation theorem which states that every f ∈L (V,F) is of the form

f (y) = ⟨y,w⟩

for a unique w ∈V.

9. ↑Let A ∈L (V,W ) where V,W are two finite dimensional inner product spaces, bothhaving field of scalars equal to F which is either R or C. Let f ∈L (V,F) be givenby

f (y)≡ ⟨Ay,z⟩

where ⟨⟩ now refers to the inner product in W. Use the above problem to verify thatthere exists a unique w ∈V such that f (y) = ⟨y,w⟩ , the inner product here being theone on V . Let A∗z≡ w. Show that A∗ ∈L (W,V ) and by construction,

⟨Ay,z⟩= ⟨y,A∗z⟩ .

In the case that V = Fn and W = Fm and A consists of multiplication on the left byan m×n matrix, give a description of A∗.

10. Let A be the linear transformation defined on the vector space of smooth functions(Those which have all derivatives) given by A f = D2 +2D+1. Find ker(A). Hint:First solve (D+1)z = 0. Then solve (D+1)y = z.

11. Let A be the linear transformation defined on the vector space of smooth functions(Those which have all derivatives) given by A f = D2 + 5D+ 4. Find ker(A). Notethat you could first find ker(D+4) where D is the differentiation operator and thenconsider ker(D+1)(D+4) = ker(A) and consider Sylvester’s theorem.

12. Suppose Ax = b has a solution where A is a linear transformation. Explain why thesolution is unique precisely when Ax = 0 has only the trivial (zero) solution.

13. Verify the linear transformation determined by the matrix(1 0 20 1 4

)

maps R3 onto R2 but the linear transformation determined by this matrix is not oneto one.

14. Let L be the linear transformation taking polynomials of degree at most three topolynomials of degree at most three given by

D2 +2D+1

where D is the differentiation operator. Find the matrix of this linear transformationrelative to the basis

{1,x,x2,x3

}. Find the matrix directly and then find the matrix

with respect to the differential operator D+1 and multiply this matrix by itself. Youshould get the same thing. Why?