Linear Algebra

Workbook

Paul Yiu

Department of Mathematics

Florida Atlantic University

Last Update: November 21

Student:

Fall 2011

Checklist

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1 2 3 4 5 6 7 8 9 10

A

B xxxC

D xxx xxx xxxE

F

F xxx xxx xxx xxx xxx xxx xxxG

H

I

J

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Problem A1.Prove that a vector space cannot be the union of two propersubspaces.

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Problem A2.Show that two subsetsA andB of a vector spaceV generatethe same subspace if and only if each vector inA is a linearcombination of vectors inB and vice versa.

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Problem A3.Show that{1, x, x2, . . . , xn, . . .} is a basis ofF [x].

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Problem A4.Let H be the quaternion algebra. Letx = a+bi+cj+dk ∈ H.Define its conjugate to be the quaternion

x = a − bi − cj − dk.

(i) Show thatxx = xx = (a2 + b2 + c2 + d2).(ii) Show thatH is a division algebra,i.e.,

xy = 0 =⇒ x = 0 or y = 0.

(ii) Let q be a nonzero quaternion. Show that{q, qi, qj, qk}is a basis ofH.

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Problem A5.Complete the following table to verify the associativity ofthequaternion algebraH.

a b c bc a(bc) ab (ab)c

i i j k ik = −j −1 −ji j ij i ii j jj i jj j i

i j ki k j

Why is this enough?

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Problem A6.(a) If X is a subspace ofV , thendimX ≤ dimV .(b) If dimV = n, the onlyn-dimensional subspace ofV is Vitself.

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Problem A7.Let A = {a1, . . . , ar} be a set of vectors such that each subsetof k vectors is linearly dependent. Prove that Span(A) is atmost(k − 1)-dimensional.

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Problem A8.Let V be a vector space overF , andA a subspace ofV . Twovectorsx, y ∈ V have two linearly independent linear combi-nations which are inA. Show thatx andy are both inA.

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Problem A9.Let F be aninfinite field. Prove that a vector space overFcannot be a finite union of proper subspaces.

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Problem A10.Let a, b, c ∈ R, not all zero, and

X = {(x, y, z) ∈ R3 : ax + by + cz = 0}.

(a) Show thatX is a subspace ofR3.(b) Find a subspaceY such thatR3 = X ⊕ Y , and deduce

thatX is 2-dimensional.

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Problem B1.What are the inverses of the elementary matricesEi,j, Ei(λ),andEi,j(λ)?

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Problem B2.Let M be ann × n matrix overF that can be brought to theidentity matrix by a sequence of elementary row operations.Show thatM is invertible, and find its inverse.

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Problem B3.Let A andB ben × n matrices such thatAB = In. Show thatBA = In.

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Problem B4.Find the inverse of then × n matrixA = (ai,j) where

ai,j =

{

1 if i ≤ j,

0, if i > j.

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Problem B5.Consider a system of linear equationsAx = b. Suppose theentries of the matricesA and b are all rational numbers, andthe system has a nontrivial solution inC. Show that it has anontrivial solution inQ.

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Problem B6.(corrected) Assume that the system of equations

cy+bz = a,

cx +az = b,

bx+ay = c

has a unique solution forx, y, z. Prove thatabc 6= 0 and findthe solution.

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Problem B7.(corrected) Givenp, q ∈ R, not both zero, consider the systemof equations

py + qz = a,

qx + pz = b,

px+ qy = c

in x, y, z.(a) Show that the system has a unique solution if and only if

p + q 6= 0. In this case, find the solution.(b) Supposep + q = 0. Show that the system has solutions if

and only ifa + b + c = 0.(c) Solve the system of equations in each of the following

cases:(i) p = q = 1;(ii) p = 1, q = −1.

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Problem B8.(a) Find the determinant of the matrix

a + 1 1 11 a + 1 11 1 a + 1

.

(b) Consider the system of equations

(a + 1)x+ y + z = p,

x +(a + 1)y + z = q,

x + y + (a + 1)z = r

in x, y, z.Solve the system in the following cases:(i) a 6= 0,−3;(ii) a = 0;(iii) a = −3.

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Problem B9.(corrected) Leta1, . . . , an be given numbers. Compute thedeterminant of then × n matrixA = (aij), where

aij = ai−1

j .

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Problem C1.Let f : X // Y be a linear transformation. Ifx1, . . . , xn

are linearly dependent, show thatf(x1), . . . ,f(xn) are linearlydependent.

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Problem C2.Show that two finite dimensional vector spaces overF are iso-morphic if and only if they have equal dimensions.

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Problem C3.(corrected) Letq = a + bi + cj + dk be a nonzero quaternion.

(a) Prove that the left multiplicationLq : H // H given byLq(x) = qx is an isomorphism.

(b) Find the matrix ofLq relative to the basis1, i, j, k ∈ H.

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Problem C4.Let s : ℓ2 // ℓ2 be the map

s(x0, x1, x2, . . . ) = (0, x0, . . . )

(a) Is this a linear map?(b) Iss a monomorphism?(c) Iss an epimorphism?

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Problem C5.(Previously mislabeled C4)Let X be a finite dimensional vector space overF , andf :X // X a linear transformation. If the rank off 2 = f ◦ f isequal to the rank off , prove that

ker f ∩ Imagef = {0}.

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Problem C6.Let f, g : X // Y be linear transformations. Prove that

rank(f + g) ≤ rank(f) + rank(g).

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Problem C7.Let V be a finite dimensional vector space overF . If f, g :V // V are linear transformations such thatg ◦ f = ιV , provethatf ◦ g = ιV .

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Problem C8.Given linear transformationsg : Y // Z andh : X // Z,prove that there exists a linear transformationf : X // Ysuch thath = g ◦ f if and only if Imageh ⊂ Imageg.

X Yf

//_____X

Z

h

��

Y

Z

g

�����������������

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Problem C9.Let X andY be vector subspaces ofV such thatV = X ⊕ Y .Prove thatV/X andY are isomorphic.

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Problem C10.Let X = X1 ⊕ X2, andV = V1 ⊕ V2 with V1 ⊂ X1 andV2 ⊂ X2. Prove thatX/V ≈ (X1/V1) ⊕ (X2/V2).

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Problem D1.Let f : X // Y be a linear transformation. Show thatf× :HomF (Y, Z) // HomF (X, Z) defined by

f×(g) = g ◦ f

is a linear transformation.

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Problem D2.(a) LetX andY be vector spaces overF . Describe the vectorspace structure (overF ) of X × Y .

(b) Givenf : V // X andg : V // Y , show that there is aunique linear maph : V // X × Y such thatπ1 ◦ h = f andπ2 ◦ h = g.

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Problem D3.Let B = {e1, e2, e3, e4} be a basis of a4-dimensional spaceVoverF , andB× = {e×

1, e×

2, e×

3, e×

4} its dual basis. Find the

dual basis ofC = {e1, e1 + e2, e1 + e2 + e3, e1 + e2 + e3 + e4}in terms ofe×

1, . . . ,e×

4.

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Problem D4.Let n be an odd number. Decide if the permutation

(

1 2 · · · k · · · n2 4 · · · 2k · · · 2n

)

is an even or odd permutation. Here in the second row, thevalues are taken modulon. (Thus,2n is replaced byn).

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Problem D5.LetM be the space of2 × 2 matrices overF , and

A =

(

a bc d

)

.

Compute the determinant of the linear transformationΦ :M // M defined by

Φ(X) = AX.

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Problem D6.Prove that the isomorphismΦX : X // X×× defined by

ΦX(x)(ϕ) = ϕ(x) for x ∈ X, ϕ ∈ X×

is natural in the sense that for every linear transformationf : X // Y , the diagram

X×× Y ××

f××//

X

X××

ΦX

��

X Yf

// Y

Y ××

ΦY

��

is commutative.

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Problem D7.Here is Puzzle 128 of Dudeney’s famous536 Curious Prob-lems and Puzzles. Take nine counters numbered 1 to 9, andplace them in a row in the natural order. It is required in asfew exchanges of pairs as possible to convert this into a squarenumber. As an example in six moves we give the following:(7846932), which give the number 139854276, which is thesquare of 11826. But it can be done in much fewer moves.

The square of 12543 can be found in four moves, as in thefirst example below. The squares of 25572, 26733, and 27273can also be obtained in four moves.

1 2 3 4 5 6 7 8 9(25) 1 5 3 4 2 6 7 8 9(37) 1 5 7 4 2 6 3 8 9(38) 1 5 7 4 2 6 8 3 9(34) 1 5 7 3 2 6 8 4 9

1 2 3 4 5 6 7 8 9

6 5 3 9 2 7 1 8 4

1 2 3 4 5 6 7 8 9

7 1 4 6 5 3 2 8 9

1 2 3 4 5 6 7 8 9

7 4 3 8 1 6 5 2 9

However, there is one, which is the square of a number of theform aabbc, which can be made inthree moves. Can you findthis number?

1 2 3 4 5 6 7 8 9

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Problem E1.1 Let A be a3× 3 matrix. Prove that the characteristic polyno-mial of A is

det(A−λI) = −λ3+(a11+a22+a33)λ2−(A11+A22+A33)λ+detA.

1Typo corrected: The constant term was previously set as− detA.

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Problem E2.Let λ be an eigenvalue ofM . Prove thatλk is an eigenvalue ofMk for every integerk ≥ 1.

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Problem E3.Let

MB(f) =

1 2 12 1 11 1 2

.

Find the eigenvalues and eigenvectors off and show that it isdiagonalizable.

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Problem E4.Let

MB(f) =

3 −1 17 −5 16 −6 2

.

Find the eigenvalues and eigenvectors off and show that it isnot diagonalizable.

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Problem E5.Diagonalize the linear operator onR3:

MB(f) =

2 2 −22 5 −4−2 −4 5

.

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Problem E6.(a) LetM be then × n matrix whose entries are all1’s. Findthe eigenvectors and eigenvalues ofM .

(b) LetM ′ be then×n matrix whose diagonal entries are allzero, and off diagonal entries all1’s. Computedet M ′.

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Problem E7.Find the eigenvalues and eigenvectors of the right shift operatorR on ℓ2:

R(x0, x1, x2, . . . ) = (0, x0, x1, . . . ).

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Problem E8.Let V be a finite dimensional vector overF , andf, g : V //Vlinear operators onV . Prove or disprove

(a) Every eigenvector ofg ◦ f is an eigenvector off ◦ g.(b) Every eigenvalue ofg ◦ f is an eigenvalue off ◦ g.

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Problem E9.Complete the details for the proof ofdimC V C = dimR V .

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Problem E10.LetV be a complex vector space, with conjugationχ : V //V .Prove that a subspaceW ⊂ V is the complexification of a realvector spaceS if and only if W is closed underχ.

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Problem F1.Prove that a positive definite inner product is nondegenerate: ifx ∈ V is such that〈x, y〉 = 0 for everyy ∈ V , thenx = 0.

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Problem F2.Prove the triangle inequality in a positive definite inner productspace: Forx, y ∈ V , ||x + y|| ≤ ||x|| + ||y||. Equality holds ifand only ifx andy are linearly dependent.

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Problem F3.(Typos corrected) Verify the polarization identity in a real innerproduct space: foru, v ∈ V ,

〈u, v〉 =1

4

(

||u + v||2 − ||u − v||2)

.

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Problem F4.Let (a, b, c) be a unit vector ofR3.

(a) Find the orthogonal projection of the vector(u, v, w) ∈R3 on the planeax + by + cz = 0.

(b) Find the matrix, relative to the standard basise1, e2, e3, ofthe orthogonal projectionπ : R3 // R3 onto the planeax +by + cz = 0.

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Problem F5.Prove that if the columns of ann × n matrix with entries inCare mutually orthogonal unit vectors, then so are the rows.

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Problem F6.Let a, b ∈ R. Prove that there are two orthogonal unit vectorsu, v ∈ R3 of the formu = (u1, u2, a) andv = (v1, v2, b) if andonly if a2 + b2 ≤ 1.

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Problem F7.Let u = (a1, b1, c1) andv = (a2, b2, c2) be nonzero vectors ofR3. Find a basis of the orthogonal complement of the span ofu andv.

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Problem F8.(corrected) In the space of continuous functions over[−1, 1]consider the inner product

〈f, g〉 :=

∫

1

−1

f(x)g(x)dx.

For eachn = 1, 2, 3, 4, find a monic polynomialpn(x) of de-green and with leading coefficient1, orthogonal to all polyno-mials of lower degrees.

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Problem F9.(new) Prove that for everyn×n matrixA, In +AtA is positivedefinite, and is nonsingular.

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Problem F10.(new) LetV be a vector space with a positive definite innerproduct, andf : V // V a mapping satisfying〈f(x), f(y)〉 =〈x, y〉 for all x, y ∈ V . Prove thatf is a linear transformation.

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Problem F11.(new) Find the angles between the two2-planes:A : y = AxandB : y = Bx, where

A =

(

0 −11 0

)

and B =

(

1 00 −1

)

.

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Problem F12.Find the equation of the2-plane inR4 containing the vector(1, 0, 0, 0) and isoclinic with the2-plane

y =

(

1 00 0

)

x.

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Problem F13.Prove thatLa = Lt

a for a quaterniona.

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Problem G1.Let π : V // V be a linear transformation satisfyingπ2 = π.Prove thatπ is the projection ofV onto Imageπ alongkerπ.

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Problem G2.A linear transformationf : V // V is called(i) an idempotent iff 2 = f ,(ii) an involution if f 2 = ιV .

Prove thatf is an idempotent if and only ifg := 2f − ι is aninvolution.

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Problem G3.Let f : V // V be a linear transformation. Prove that the in-duced mapf∗ : Hom(V, V ) // Hom(V, V ) defined byf∗(g) =f ◦ g is a projection if and only iff is a projection.

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Problem G4.Let f : V // V be a given linear transformation. A subspaceX ⊂ V is invariant underf if f(X) ⊂ X.

(i) Let X be a subspace invariant underf . Prove that forevery projectionπ = πX,Y , π ◦ f ◦ π = f ◦ π.

(ii) Suppose the relation in (i) holds for some direct sum de-compositionV = X ⊕ Y . Prove thatX is invariant underf .

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Problem G5.Prove that the Riesz mapR : V × // V defined by

〈R(f), y〉 = f(y) for everyy ∈ V,

is an isomorphism.

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Problem G6.Let f : X //Y be a linear transformation between inner prod-uct spaces. Show thatf ∗∗ = f .

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Problem G7.Without making use of the Theorem in§7B, prove that iff :V //V is a self adjoint linear operator, there is an orthonormalbasis ofV consisting of eigenvectors off .

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Problem G8.Let e1 = (1, 0) ∈ C2 regarded as a Hermitian space. Find allunit vectors inC2 orthogonal toe1.

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Problem G9.Prove the hyperplane reflection formula

τv(x) = −x + 2〈x, v, v〉.

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Problem G10.Given a unit vectorv = (a, b, c) ∈ R3, find the matrix, rela-tive to the standard basis ofR3, of the rotation aboutv by 180degrees.

An n × n matrix overC is(i) Hermitian if A∗ = A,(ii) skew-Hermitian ifA∗ = −A,(iii) unitary if A∗A = I = AA∗.

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Problem H1.

Let A =

(

1 −11 1

)

.

(a) Show thatA is normal.(b) IsA Hermitian? skew-Hermitian? or unitary?

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Problem H2.Let U be a unitary matrix.

(a) LetA be a normal matrix.U ∗AU is normal.(b) Let A be a Hermitian matrix. Show thatU ∗AU is Her-

mitian.(c) LetA be a unitary matrix. Show thatAU is unitary.

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Problem H3.Let A be a diagonal matrix.

(a) Show thatA is Hermitian if and only if its entries are real.(b) Show thatA is skew-Hermitian if and only if its diagonal

entries are purely imaginary.(c) Show thatA is unitary if and only if its diagonal entries

are unit complex numbers.

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Problem H4.Prove that the trace of a matrix is a similarity invariant: ifA, P ∈ Mn(F ) andP nonsingular, then Tr(P−1AP ) = Tr(A).

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Problem H5.LetA = (ai,j) andB = (bi,j) be complexn×n matrices. Provethat if A and B are unitarily equivalent, then

∑

i,j |ai,j|2 =

∑

i,j |bi,j|2.

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Problem H6.Let A = (ai,j) be a complex matrix with eigenvaluesλ1, . . . ,λn. Prove that ifA is normal, then

∑

i,j |ai,j|2 =

∑

i |λi|2.

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Problem H7.Show that the matrices

A =

1 3 00 2 40 0 3

and B =

1 0 00 2 50 0 3

are similar and satisfy∑

i,j |ai,j|2 =

∑

i,j |bi,j|2, but are not

unitarily equivalent.

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Problem H8.Let A andB be similar2 × 2 matrices satisfying

∑

i,j |ai,j|2 =

∑

i,j |bi,j|2. Prove thatA andB are unitarily equivalent.

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Problem H9.Let A be a nonsingular matrix. Prove that every matrix whichcommutes withA also commutes withA−1.

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Problem H10.Let A be ann × n matrix for whichAk = 0 for somek > n.Prove thatAn = 0.

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Problem I1.A linear transformationf : R3 // R3 has matrix

1

3

2 2 11 −2 22 −1 −2

relative to the standard basise1, e2, e3. Show thatf is anisometry.

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Problem I2.Find the invariant1-planes and invariant2-planes of the isom-etryf in Problem I1 with matrix

1

3

2 2 11 −2 22 −1 −2

relative to the standard basise1, e2, e3.

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Problem I3.Let f be an isometry ofR4 with matrix

cos θ sin θ− sin θ cos θ

cos θ sin θ− sin θ cos θ

,

with 0 < θ < 2π relative to the standard basise1, e2, e3, e4.Show thatf rotates every vector ofR4 by an angleθ.

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Problem I4.Let r ands be orthogonal pure unit quaternions.

(a) Show thatr, s, rs is an orthonormal basis of the3-planei, j, k.

(b) Show that the solution space of the equationru = ur isthe span of1 andr.

(c) Show that the solution space of the equationru = −ur isthe orthogonal complement of Span(1, r).

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Problem I5.Let a ∈ H be a fixed unit quaternion. Consider the isometryfa : H // H defined by

f(u) = aua.

(a) Prove thatf leaves the3-planeW := Span(i, j, k) in-variant.

(b) Prove thatf induces an orientation preserving isometryonW .

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Problem I6.Use the Cayley-Hamilton theorem to find the inverse of a non-singular2 × 2 matrix

(

a bc d

)

.

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Problem I7.ComputeA10 for the matrix

A =

3 1 12 4 2−1 −1 1

.

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Problem I8.Find a real matrixB such that

B2 =

2 0 00 2 00 −1 1

.

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Problem I9.Prove or disprove: for any2 × 2 matrix A over C, there is a2 × 2 matrixB such thatA = B2.

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Problem I10.Forx ∈ R, let

Ax :=

x 1 1 11 x 1 11 1 x 11 1 1 x

.

(a) Prove thatdet(Ax) = (x − 1)3(x + 3).(b) Prove that ifx 6= 1,−3, then

A−1

x =1

(x − 1)(x + 3)A−x−2.

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Problem J1.

Find the Jordan canonical form of the all-one matrix of ordern, i.e., A = (ai,j) with ai,j = 1 for all i, j = 1, . . . , n), giventhat its eigenvalues aren (multiplicity 1) and0 (multiplicityn − 1).

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Problem J2.A 6 × 6 matrix A has eigenvalueλ of multiplicity 6, and theeigenspaceVλ has dimension3. What are the possible Jordoncanonical forms?

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Problem J3.Determine all possible Jordan canonical forms for a matrixwith characteristic polynomial(λ − 3)6 and minimal polyno-mial (λ − 3)2.

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Problem J4.What are the possible Jordan canonical forms for a matrix oforder6 whose characteristic polynomial is(λ − 2)4(λ + 3)2?

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Problem J5.Find the canonical form of the matrix

1 1 1 1 1 10 1 0 0 0 10 0 1 0 0 10 0 0 1 0 10 0 0 0 1 10 0 0 0 0 1

.

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Problem J6.Find a nonsingular matrixP such that, for the matrix

A =

0 1 1 0 10 0 1 1 10 0 0 0 00 0 0 0 00 0 0 0 0

,

P−1AP is a Jordan canonical form.

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Problem J7.Let T be an upper triangular matrix overR which commuteswith its transpose. Show thatT is a diagonal matrix.

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Problem J8.

Show that ifA =

(

λ 10λ

)

, then

An =

(

λn nλn−1

0 λn

)

for every positive integern.

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Problem J9.Let A be a2 × 2 complex matrix. Show that the series

I + A + A2 + · · · + An + · · ·

converges if and only if|λ| < 1 for each eigenvalueλ of A.

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Problem J10.Let A andB be real2 × 2 matrices such thatA2 = B2 = I,andAB + BA = 0. Prove that there exists a real nonsingularmatrixP such that

P−1AP =

(

1 00 −1

)

and P−1BP =

(

0 11 0

)

.