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9.E: Exercises for Chapter 9

( \newcommand{\kernel}{\mathrm{null}\,}\)

Calculational Exercises

1. Let (e1,e2,e3) be the canonical basis of R3 , and define
f1=e1+e2+e3,         f2=e2+e3,         f3=e3.
(a) Apply the Gram-Schmidt process to the basis (f1,f2,f3).
(b) What do you obtain if you instead applied the Gram-Schmidt process to the basis (f3,f2,f1)?

2. Let C[π,π]={f:[π,π]Rf is continuous} denote the inner product space of continuous real-valued functions defined on the interval [π,π]R, with inner product given by

f,g=ππf(x)g(x)dx, for every f,gC[π,π].

Then, given any positive integer nZ+, verify that the set of vectors

{12π,sin(x)π,sin(2x)π,,sin(nx)π,cos(x)π,cos(2x)π,,cos(nx)π} is orthonormal.

3. Let R2[x] denote the inner product space of polynomials over R having degree at most two, with inner product given by

f,g=10f(x)g(x)dx, for every f,gR2[x].

Apply the Gram-Schmidt procedure to the standard basis {1,x,x2} for R2[x] in order to produce an orthonormal basis for R2[x] .

4. Let v1,v2,v3R3 be given by v1=(1,2,1),v2=(1,2,1), and v3=(1,2,1).
Apply the Gram-Schmidt procedure to the basis (v1,v2,v3) of R3 , and call the resulting orthonormal basis (u1,u2,u3).

5. Let PR3 be the plane containing 0 perpendicular to the vector (1,1,1). Using the standard norm, calculate the distance of the point (1,2,3) to P .

6. Give an orthonormal basis for null(T), where TL(C4) is the map with canonical matrix

(1111111111111111)

Proof-Writing Exercises

1. Let V be a finite-dimensional inner product space over F. Given any vectors u,vV , prove that the following two statements are equivalent:

(a)u,v=0

(b)uu+αv for every αF.

2. Let nZ+ be a positive integer, and let a1,,an,b1,,bnR be any collection of 2n real numbers. Prove that

(nk=1akbk)2(nk=1ka2k)(nk=1b2kk)

3. Prove or disprove the following claim:
Claim. There is an inner product , on R2 whose associated norm is given by the formula
(x1,x2)=|x1|+|x2|
for every vector (x1,x2)R2 , where || denotes the absolute value function on R.


4. Let V be a finite-dimensional inner product space over R. Given u,vV, prove that
u,v=u+v2uv24

5. Let V be a finite-dimensional inner product space over C. Given u,vV , prove that

u,v=u+v2uv24+u+iv2uiv24i.

6. Let V be a finite-dimensional inner product space over F, and let U be a subspace of V. Prove that the orthogonal complement U of U with respect to the inner product , on V satisfies

dim(U)=dim(V)dim(U).

7. Let V be a finite-dimensional inner product space over F, and let U be a subspace of V. Prove that U=V if and only if the orthogonal complement U of U with respect to the inner product , on V satisfies U={0}.

8. Let V be a finite-dimensional inner product space over F, and suppose that PL(V) is a linear operator on V having the following two properties:

(a) Given any vector vV,P(P(v))=P(v). I.e., P2=P.

(b) Given any vector unull(P) and any vector vrange(P),u,v=0.

Prove that P is an orthogonal projection.

9. Prove or give a counterexample: For any n1 and ACn×n, one has

null(A)=(range(A)).

10. Prove or give a counterexample: The Gram-Schmidt process applied to an an orthonormal list of vectors reproduces that list unchanged.


This page titled 9.E: Exercises for Chapter 9 is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

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