Linear Algebra Exercises

8/13/2019 Linear Algebra Exercises

http://slidepdf.com/reader/full/linear-algebra-exercises 1/32

Chapter 1

Matrices

1.1 Consider the matrices:

A =

1 −1 0 1

2 1 1 0

−1 1 3 1

, B =

3 0 0

0 2 0

0 0 1

, C =

1

−1

2

, D =

−3 1 4 1

E =

2

, F =

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

, G =

1 4

2 5

3 6

, H =

0 0 0

1 0 0

2 4 0

e I =

1 0

0 1

.

Indicate which of these matrices are:

(a) Square matrices.

(b) Lower triangular matrices.

(c) Diagonal matrices.

(d) Scalar matrices.

1.3 Consider the real matrices

A =

3 1 0

1 1 −1

, B =

1 0 4

−1 2 −1

e C =

0 0 1

−2 −2 1

.

1



Compute:

(a) (A + B) + C ;

(b) A + (C + B);

(c) (2B + 2A) + 2C ;

(d) A − B.

1.4 Let A, B ∈ M3×3(R).

A =

1 0 0

0 1 0

0 0 1

and B =

1 1 1

1 1 1

1 1 1

,

Find X ∈ M3×3(R), such that X + A = 2(X − B).

1.5 Let

A =

1 2 −1

∈ M1×3(R) and B =

0

1

3

∈ M3×1(R).

Compute if possible, AB and B A.

1.7 Let

A =

1 2

, B =

2 1

0 2

, C =

1 −1

0 1

2 0

and D =

−1 1 1

1 −1 0

.

Compute, if possible, each one of the following products:

(a) AB.

(b) BA.

(c) CD.

(d) DC .

1.8 Consider the matrices

A =

4 2

2 1

, B =

−1 −1

2 2

, C =

0 −3

3 0

∈ M2×2(R).

Verify that:

(a) AB = BA.

(b) AB = 0 with A = 0 and B = 0.

(c) BA = C A and A = 0 but B = C .

2



1.9 Let D, D ∈ Mn×n(K) be diagonal matrices. Prove that DD is a diagonal matrix with

(DD)ii = diidii, i = 1, . . . , n and conclude that DD = DD.

1.10 Prove the following statements:

(a) If A ∈ Mm×n(K) has the i row null than, for any matrix B ∈ Mn× p(K), the

matrix AB has the i row null.

(b) If B ∈ Mn× p(K) has the k column null than, for any matrix A ∈ Mm×n(K), the

matrix AB has the k column null.

(c) If A ∈ Mm×n(K) has the row i equal to the row j , with i = j , than, for any

matrix B ∈ Mn× p(K), the matrix AB has the row i equal to the row j.

(d) If B ∈ Mn× p(K) has the column k equal to the column l, with k = l, than, for

any matrix A ∈ Mm×n(K), the matrix AB has the column k equal to the column

l.

1.15 Let D ∈ Mn×n(K) be a diagonal matrix. Find Dk for k ∈ N0.

1.19 Let A ∈ Mn×n(K). Use the exercice 1.10, to prove that:

(a) If A has a null column than A is not invertible.

(b) If A has the column i equal to the column j , with i = j , than A is not invertible.

1.26 Let A ∈ M3×3(R) be an invertible matrix with A−1 =

1 1 2

0 1 3

4 2 1

.

(a) Find a matrix B such that AB =

1 2

0 1

4 1

∈ M3×2(R). Justify that B is the only

matrix that verifies the equality.

(b) Find a matrix C such that AC = A + 2I 3. Justify that C is the only matrix that

verifies the equality.

1.34 Determine whether the matrix

A =

0 0

0 0

, B =

1 2

2 3

0 0

, C =

1 2 3

2 0 4

3 4 5

,

D =

1 2 −3

−2 0 4

3 −4 −1

, E =

0 2 3

−2 0 4

−3 −4 0

e F =

0 0 0

0 0 0

(a) is symmetric.

(b) is skew-symmetric.

3



1.42 Determine whether the matrix is an elementary matrix and if so indicate its type.

(a)

1 0 0

0 1 −1

0 0 1

.

(b)

2 0 0

0 1 0

0 0 1

.

(c)

0 1 0

0 0 1

1 0 0

.

(d)

1 0 0

0 1 0

0 0 0

.

(e) I n.

1.43 Let A ∈ M3×5(K). In each part indicate an elementary matrix E that perform the

stated row operation by multipling A on the left by E .

(a) Interchange rows 1 and 3.

(b) Multiply the row 1 by 6.

(c) Add, to the row 3, the product of the row 2 by 1

5.

1.44 By inspection indicate the product of:

(a)

0 1 0

1 0 0

0 0 1

a b c d

e f g h

i j k l

.

(b)

5 0 0

0 1 0

0 0 1

0 1 0

1 0 0

0 0 1

a b c d

e f g h

i j k l

.

(c)

a b c d

e f g h

i j k l

1 0 0 0

0 1 0 0

0 0 1 3

0 0 0 1

.

(d)

2 0

0 1

a b c

d e f

1 0 0

0 1 −5

0 0 1

.

1.46 Find the inverse of each of following elementary matrices:

(a)

1 0 0

0 5 0

0 0 1

(b)

0 0 1

0 1 0

1 0 0

(c)

1 0 0

0 1 0

−3 0 1

4



1.48 Determine whether the matrix is in the row-echelon form.

(a) I n.

(b)

0 0 0

5 1 4

0 1 3

0 0 2

.

(c)

0 5 0 0

.

(d)

0 1 0

0 0 1

0 0 1

.

1.49 Indicate a row-echelon form matrix that is row-equivalent to each of the following

matrices:

(a)

1 2 1

2 1 0

−1 0 1

.

(b)

2 4 −2 6 0

4 8 −4 7 5

−2 −4 2 −1 −5

.

(c)

2 2 1

−2 −2 1

1 1 2

.

1.51 Determine whether the matrix is in the reduced row-echelon form.

(a)

0 0 0 1 5

.

(b)

0 1 0 1 1

0 0 1 1 1

0 0 0 0 0

.

(c)

0 1 2 5 0

0 0 0 1 1

0 0 0 0 0

.

(d) 0 1 2 5

.

(e)

1

0

0

.

1.52 Indicate the reduced row-echelon form for each of the following matrices:

(a)

1 2 1

2 1 0

−1 0 1

.

(b)

2 4 −2 6 0

4 8 −4 7 5

−2 −4 2 −1 −5

.

(c)

2 2 1

−2 −2 1

1 1 2

.

5




A1 =

0 1 0 1 1

2 3 0 2 1

−2 −1 0 1 −1

, A2 =

2 1 0

1 1 2

3 −1 1

−1 1 0

,

A3 =

1 4 2

2 3 1

−1 1 1

e A4 =

2 1 1

0 0 1

1 −1 2

.

Find the rank of Ai, with i = 1, 2, 3, 4.

1.58 Find the rank of the following matrices for each α, β ∈ R.

Aα =

1 0 −1 1

1 1 0 1

α 1 −1 2

, Bα =

1 −1 0 1

1 1 0 −1

α 1 1 0

0 1 α 1

,

C α,β =

0 0 α

0 β 2

3 0 1

e Dα,β =

α 0 −1 β

1 0 β 0

1 1 1 1

1 1 0 1

.

1.59 Compute the rank of the following matrices and justify that they aren’t row-equivalent.

1 2

4 8

e

0 1

1 2

1.62

(a) Compute the set of values α ∈ R, such that the matrix

1 2 0

1 4 2

2 4 5 + α

∈ M3×3(R)

is invertible.

(b) Compute the set of values α and the set of values β , with α, β ∈ R, such that the

matrix

1 2 1

1 α + 3 2

2 4 β

∈ M3×3(R)

is invertible.

1.65 Let A = 1 −1

2 0

∈ M2×2(R

).

(a) Show that A is invertibleand compute A−1.

(b) Write A−1 and A as a product of elementary matrices.

6




A =

3 1 0

1 2 1

2 −1 −1

∈ M3×3(R), B =

1 −1 0

2 1 2

0 1 −1

∈ M3×3(R),

C =

1 1 + i

−i 1

∈ M2×2(C) e D =

1 −1 1 2

2 −2 1 1

1 −1 0 1

−2 0 2 −2

∈ M4×4(R).

Determine whether the matrix is invertible; if so find the inverse.

7





Chapter 2

Systems of Linear Equations

2.2 Let

A =

1 1 2 −1

2 2 −2 2

0 0 6 −4

∈ M3×4(R) B =

−1

4

−6

∈ M3×1(R)

and (S ) the system of linear equations AX = B. Without solving the system show

that:

(a) (−1, 1, 1, 3) is a solution of (S ).

(b) (1, 0, 1, 0) is not a solution of (S ).

2.3 Justify that there exists a system of linear equations, (S ) : AX = B , with

A =

1 0 −1

2 4 3

−1 0 2

3 4 2

∈ M4×3(R)

such that (1, 2, 3) is a solution of (S ). Indicate the equations of a system in these

conditions.

9



2.7 With the information that you have in the next table find, if possible, if each of the

systems of linear equations AX = B is consistent (with exactly one solution or with

infinitely many solutions) or inconsistent. For the systems that are consistent computethe number of free variables.

Matrix A r(A) r([A | B ])

(a) 3×3 3 3

(b) 3×3 2 3

(c) 3×3 1 1

(d) 5×7 3 3

(e) 5×7 2 3

(f) 6×2 2 2

(g) 4×4 0 0

2.8 Find a consistent system of linear equations with 3 unknowns which has the following

number of free variables:

(a) 1.

(b) 2.

Can be 3 the number of free variables?

2.11 Consider the following system of linear equations with variables x, y and real constants. x − y = 1

3x − 3y = k

Find the set C of real values k for which the system is

(a) inconsistent.

(b) consistent with exactly one solution.

(c) consistent with infinitely many solutions.

2.15 Show that the matrix

A =

−3 2 −1

2 0 −2

−1 1 1

∈ M3×3(R)

is invertible. Use A−1 to compute the solution of the linear system with unknowns

x,y,z, and real constants,

−3x + 2y − z = α

2x − 2z = β

−x + y + z = γ

, with α, β , γ ∈ R.

10



2.35 For each α ∈ R and for each β ∈ R, consider the system of linear equation with

unknowns x, y,z, and real constants,

x + y − z = 1

−x − αy + z = −1

−x − y + (α + 1)z = β − 2

.

(a) Find for what values of α and β the system is inconsistent, consistent with one

or infinitely many solutions. Considering the values for which the system is con-

sistent, indicate the number of free variables.

(b) Find the set of solutions when α = 0 and β = 1.

2.37 For each α ∈ R and for each β ∈ R, consider the system of linear equation with

unknowns x, y,z, and real constants,

(S α,β)

x + αy + βz = 1

α(β − 1)y = α

x + αy + z = β 2

.

(a) Find for what values of α and β the system is inconsistent, consistent with one or

infinitely many solutions. Considering the values for which the system is consistentindicate the number of free variables.

(b) i. Justify that S 2,2 has only one solution.

ii. Justify that the coefficient matrix of S 2,2 is invertible.

iii. Compute the solution of S 2,2, using the inverse of the coefficient matrix of

the system.

11





Chapter 3

Determinants

3.1 Compute the determinant of the following matrices:

(a) A =

1 1

1 2

∈ M2×2(R).

(b) B =

1 2

2 1

∈ M2×2(R).

(c) C =

1 i

i −1

∈ M2×2(C).

3.2 Let A =

0 a a2

a−1 0 a

a−2 a−1 0

∈ M3×3(R), with a = 0. Compute the determinant of A

using the Sarrus Rule.

3.3 Let A =

1 0 3

−1 2 4

3 1 2

∈ M3×3(R). Compute:

(a)

a11.

(b) a32.

(c) a23.

3.4 Compute using two different ways the determinant of each of the following matrices:

(a) A =

1 1 0

2 1 1

1 1 1

∈ M3×3(R).

(b) B =

1 0 i

0 0 2

−i 2 1

∈ M3×3(C).

(c) C =

1 0 −1 0

−2 0 2 −1

1 1 −1 1

3 3 −6 6

∈ M4×4(R).

13



3.6 For each λ ∈ R, consider

Aλ =

3 − λ −3 2

0 −

2−

λ 20 −3 3 − λ

.

Indicate the set of values of λ for which det Aλ = 0.

3.10 Let A =

a b c

d e f

g h i

∈ M3×3(R) such that det A = γ .

Indicate using γ , the value of each of the following determinants:

(a)

d e f

g h i

a b c

.

(b)

3a 3b 3c

−d −e −f

4g 4h 4i

.

(c)

a + g b + h c + i

d e f

g h i

.

(d)

−3a −3b −3c

d e f

g − 4d h − 4e i − 4f

.

(e)

b e h

a d g

c f i

.

3.15 For each k ∈ R, consider the matrix

Bk =

1 0 −1 0

2 −1 −1 k

0 k −k k

−1 1 1 2

∈ M4×4(R).

Indicate the set of values of k for which we have det Bk = 2.

3.19 For each t ∈ R, let

At =

1 t −1

2 4 −2

−3 −7 t + 3

∈ M3×3(R).

Indicate the set of values of t for which At is invertible.

3.20 Let A, B , C ∈ Mn×n(R) such that det A = 2, det B = −5 and det C = 4. Compute

det(ABC ), det (3B) and det

B2C

.

3.21 Show that, for all A, B ∈ Mn×n(K), we have:

(a) det(AB) = det (BA).

(b) If AB is an invertible matrix then A and B are invertible matrices.

14



3.22 Let A = −1 2

∈ M1×2(R) e B =

0

1

∈ M2×1(R).

(a) Show that det (AB) = det (BA).

(b) What can you say when you observe this exercise and the exercise 3.21 (a)?

3.23 Let A ∈ Mn×n(K) an idempotent matrix (that is, A2 = A). Show that det A ∈ {0, 1}.

3.25 Consider the following matrices:

A =

1 −1 1 1

0 2 4 4

1 3 1 1

0 0 −2 0

∈ M4×4(R) e B =

2 1 −1

1 1 1

−1 0 2

∈ M3×3(R).

(a) Compute det A and det B.

(b) Indicate if one of the matrices above are invertible. For each one of the invertible

matrices indicate the determinant of its inverse.

(c) Indicate if the following systems have one solution or many infinitely solutions.

i. AX = 0.

ii. BX = 0.

3.26 Let

A =

−4 −3 −3

1 0 1

4 4 3

∈ M3×3(R)

Verify that adj A = A.

3.28 Show that each of the following matrices are invertible and compute its inverse using

its adjoint matrix.

(a) A =

3 1 2

1 2 1

2 2 2

∈ M3×3(R).

(b) V α =

cos α − sen α

sen α cos α

, com α ∈ R.

(c) A =

z w

−w z

∈ M2×2(C), com z = 0 or w = 0.

3.32 Let

A =

1 2 3

0 2 1

1 1 1

∈ M3×3(R), B =

14

7

6

∈ M3×1(R)

and let (S ) be the linear system AX = B .

(a) Compute det A and justify that the system (S ) is a Cramer system.

(b) Using the Cramer Rule, compute the solution of the system (S ).

15



3.33 For each k ∈ R, consider the matrix

Ak =

1 −k 1

0 k kk k −k

∈ M3×3(

R).

(a) Using determinants, indicate the values of k for which the matrix Ak is invertible.

(b) For k = −1 justify that the linear system

AX =

1

0

0

is a Cramer system and compute its solution.

16



Chapter 4

Vector spaces

4.3 Consider E = R2 and define addition and scalar multiplication as follows:

(a1, a2) + (b1, b2) = (a1 + b1, a2 + b2)

α(a1, a2) = (αa1, 0),

for all (a1, a2), (b1, b2) ∈ R2 and α ∈ R. Prove that (R2, +, ·) is not a real vector space.

4.6 For each of the following vector spaces indicate the element 0E :

(a) E = R4.

(b) E = M2×3(R).

(c) E = R3[x].

4.8 Let E be a vector space over K. Let α, β ∈ K and u, v ∈ E . Justify that:

(a) If αu = αv and α = 0K then u = v.

(b) If αu = βu and u = 0E then α = β .

4.13 Determine which of the following sets are subspaces of the corresponding vector space

.

(a) F 1 =

(a, b) ∈ R2 : a ≥ 0

in R2.

(b) F 2 = {(0, 0, 0), (0, 1, 0), (0, −1, 0)} in R3.

(c) F 3 =

(a,b,c) ∈ R3 : 2a = b ∧ c = 0

in R3.

(d) F 4 = (a,b,c) ∈ R

3

: 2a = b in R3

.

17



4.15 Show that each of the following set of matrices are a vector space of Mn×n(K):

(a) With all diagonal elements equal to zero.

(b) Upper triangular.

(c) Diagonal.

(d) Scalar.

(e) Symmetric.

(f) Semi-symmetric.

4.16 Justify that each of the following set of matrices are not a vector space of Mn×n(K)

(a) With, at least one diagonal element different from zero.

(b) Invertible.

(c) Not invertible.

4.20 Show that each of the following sets are subspaces of the corresponding vector space.

(a) F =

(a,b,c,d) ∈ R4 : a − 2b = 0 ∧ b + c = 0

in R4.

(b) G =

a b

c d

∈ M2×2(R) : a − 2b = 0 ∧ b + c = 0

in M2×2(R).

(c) H = ax3

+ bx2

+ cx + d ∈R

3[x] : a − 2b = 0 ∧ b + c = 0 in R

3[x].

4.22 Let G =

a a + b

−b 0

: a, b ∈ R

. By indicating a spanning set of G, show that G is

a subspace of M2×2(R).

4.23 Show that each of the following sets are subspaces of the corresponding vector space,

presenting a span sequence for each of them.

(a)

(a,b,c) ∈ R3 : a − c = 0

em R3.

(b) a b

c d ∈ M2×2(R) : a + d = 0 em M2×2(R).

(c)

ax3 + bx2 + cx + d ∈ R3[x] : a − 2c + d = 0

em R3[x].

4.31 Let E be a vector space over K and let u1, u2, u3 ∈ E . Justify the statements:

(a) S = (u1, u2, u3) is linearly independent if, and only if,

S = (u1, u1 + u2, u1 + u2 + u3)

is linearly independent.

(b) S = (u1, u2, u3) is linearly independent if, and only if,

S = (u1 − u2, u2 − u3, u1 + u3)


18



(c) The sequence

S = (u1 − u2, u2 − u3, u1 − u3)


4.33 Consider the subspace of R3 F =(2, 3, 3)

. Determine two distinct bases for F .

4.35 Let G =

a a + b

−b 0

: a, b ∈ R

the subspace of M2×2(R) (see exercise 4.22). Find

a basis of G.

4.41 Consider the subspace of R3,

F =(1, 2, 1), (2, −1, −3), (0, 1, 1)

.

(a) Verify that(1, 2, 1), (2, −1, −3), (0, 1, 1)

is not a basis for F .

(b) Find a subsequence of the previous sequence that is a basis for F .

4.44 Consider in M2×2(R), the bases

B =

1 0

0 0

,

1 1

0 0

,

1 1

1 0

,

1 1

1 1

e

B =

1 0

0 0

,

0 1

0 0

,

0 0

1 0

,

0 0

0 1

.

(a) Find the coordinate sequence of the vector

4 3

2 1

relative to bases B and B .

(b) Find the coordinate sequence of an arbitrary vector a b

c d

∈ M2×2(R) relative to bases B e B .

4.48 Consider the following subspaces of R4

F =

(a,b,c,d) ∈ R

4 : a − c = 0 ∧ a − b + d = 0

e

G =(1, 1, 0, 1), (2, 1, 2, −1)

.

Find a basis for F ∩ G.

4.52 Consider the subspaces of R4,

F =

(a,b,c,d) ∈ R4 : a − b = 0 ∧ a = b + d

,

G =

(a,b,c,d) ∈ R

4 : b − c = 0 ∧ d = 0

and

H =(1, 0, 0, 3), (2, 0, 0, 1)

.

Find a basis for

19



(a) F .

(b) G.

(c) F + G.

(d) F + H .

4.56 Consider the subspaces of R2,

F =

(x, y) ∈ R2 : y = 0

,

G =

(x, y) ∈ R2 : x = 0

and

H = (x, y) ∈ R2

: x = y .

Justify that

F ⊕ G = R2 = F ⊕ H.

4.62 Consider the subspaces of M2×2(R),

F =

a b

0 0

: a, b ∈ R

and G =

0 0

c d

: c, d ∈ R

.

(a) Show that M2×2(R) = F ⊕ G.

(b) Considering A =

4 5

0 6

determine the projection of A onto F , along G, and the

projection of A onto G, along F .

4.65 Consider the sequences of vectors of R3,

S k =(1, 0, 2), (−1, 2, −3), (−1, 4, k)

.

Find the set of values of k for which S k is a basis for R3.


F = (1, 0, 1, 0), (−1, 1, 0, 1), (1, 1, 2, 1) .

(a) Find a basis for F .

(b) Verify that (1, 2, 3, 2) ∈ F .

(c) Find a basis for R4 for which the basis indicated in (a) is a subset.

4.71 Consider the sequences of vectors of M3×1(R),

S 1 = 1

−11

,

1

10

and S 2 = 1

−11

,

1

10

,

2

01

.

Determine if S i, i = 1, 2, is a linearly dependent sequence of vectors, in that case find a

vector of the sequence that is a linear combination of the others vectors of the sequence.

20



4.74 Indicate the dimension and a basis for each of the following subspaces:

(a) F =

2x3 + 2x2 − 2x, x3 + 2x2 − x − 1, x3 + x + 5,

x3 + 3, 2x3 + 2x2 − x + 2 of R3[x].

(b) G =

1 1

1 1

,

1 1

1 0

,

2 −3

1 1

,

4 −1

3 2

of M2×2(R).

4.155 Consider the subspace of R4[x],

F =

a0 + a1x + a2x2 + a3x3 + a4x4 ∈ R4[x] :

−2a0 + 2a1 + a4 = 0 ∧ −a0 + a1 + 5a4 = 0} .

(a) Find a basis for F .

(b) Find a basis for R4[x] for which the basis indicated in (a) is a subsequence.

(c) If exists indicate a subspace G of R4[x] such that

dim(F + G) = 4 e dim(F ∩ G) = 1.

4.156 For each α ∈ R, consider the set:

F α = (x,y,z) ∈ R3 : x = αy ∧ αy = αz .

(a) Show that, for all α ∈ R, F α is a subspace of R3.

(b) Determine for each α, a basis for F α.

(c) Let G =(1, 1, 0), (0, 0, 2)

.

i. Determine for each α, dim(G + F α).

ii. Determine for each α, a basis for G + F α.

21





Chapter 5

Linear Transformations

5.2 Determine whether the following functions, f : R3 −→ R2, are a linear transformation.

(a) f (x , y , z) = (y, 0).

(b) f (x , y , z) = (x − 1, y).

(c) f (x , y , z) = (xy, 0).

(d) f (x , y , z) = (x, |z|).

5.4 Let g : Mm×n(K) −→ Mn×m(K) be a function such that, for all A ∈ Mm×n(K),

g(A) = A.

Justify that g is a linear transformation.

5.6 Justify that if f : R3 −→ M2×2(R) is a linear transformation then:

(a) f (0, 0, 0) =

0 0

0 0

.

(b) f (2, 4, −2) = 2f (1, 2, −1).(c) f (−3, 1, 2) = f (−2, 0, 1) + f (−1, 1, 1).

5.7 Determine whether the following functions are a linear transformations.

(a) f : R3 −→ R2 such that

f (a,b,c) = (2a, b + 1),

for all (a,b,c) ∈ R3.

(b) g : R2[x] −→ M2×2(R) such that

g(ax2 + bx + c) =

c b

a + b 2

,

for all ax2 + bx + c ∈ R2[x].

23



5.10 Find the kernel, a basis for the kernel an a basis for the range (Image space) for each

of the following linear transformations.


f (x,y,z) = (y, z),

for all (x,y,z) ∈ R3.

(b) g : M2×2(R) −→ R3 such that

g

a b

c d

= (2a, c + d, 0),

for all a b

c d ∈ M2×2(R).

(c) h : R3 −→ R2[x] such that

h(a,b,c) = (a + b)x2 + c,


(d) t : R3[x] −→ M2×2(R) such that

t(ax3 + bx2 + cx + d) =

a − c 0

0 b + d

,

for all ax3 + bx2 + cx + d ∈ R3[x].

5.14 Determine if each of the following linear transformations are a injection computing

each kernel.


f (a,b,c) = (2a, b + c, b − c),


(b) g : R2[x] −→ M2×2(R) such that

g(ax2 + bx + c) =

2a b + c

0 a + b − c

,

for all ax2 + bx + c ∈ R2[x].

5.17 Compute the nullity of the following linear transformations:

(a) f : R5 −→ R8 with dim Im f = 4.

(b) g : R3[x] −→ R3[x] with dim Im g = 1.

(c) h : R6 −→ R3 with h an onto mapping.

(d) t : M3×3(R) −→ M3×3(R) with t an onto mapping.

24



5.18 Justify that does not exists any linear transformation f : R7 −→ R3 whose kernel has

dimension less or equal to 3.

5.19 Let f : R5 −→ R3 be a linear transformation with nullity n(f ) and rank r(f ). Find all

the possible pairs (n(f ), r(f )).

5.22 Using a suitable proposition indicate if each of the following linear transformations are

a bijection.


f (a,b,c) = (2a, b + c, b − c),


(b) g : M2×2(R) −→ R3[x] such that

g

a b

c d

= (a + d)x3 + 2ax2 + (b − c)x + (a + c),

for all

a b

c d

∈ M2×2(R).

5.26 In the following cases indicate if there are a linear transformation in the conditions

given. If that is possible give an example.


Im f =(1, 0, 0, 1), (0, 1, 1, 0), (0, 1, 2, 0)

e dim Ker f = 2.

(b) g : R4 −→ R3 such that

Ker g =(0, 1, 1, 0), (1, 1, 1, 1)

e (1, 1, 1) ∈ Im g.

(c) h : R3 −→ R4 such that

Im h =(1, 2, 0, −4), (2, 0, −1, −3)

.

5.36 Consider the linear transformation: f : R3 −→ R2 such that, for all (x,y,z) ∈ R3 is

f (x,y,z) = (y, z),

and the linear transformation that, for all

a b

c d

∈ M2×2(R) is g : M2×2(R) −→ R3

such that

g

a b

c d

= (2a, c + d, 0),

.

25



(a) Compute M(f ; B , B ) considering

B =

(1, 2, 3), (0, −2, 1), (0, 0, 3)

and B =

(0, −2), (−1, 0)

.

(b) Compute M(g; B , b. c.R3) considering

B =

1 1

0 1

,

0 2

3 1

,

0 0

2 −1

,

0 0

1 0

.

5.42 Let f : R3 −→ R2 be the linear transformation such that, for all (a,b,c) ∈ R3, is

f (a,b,c) = (a + b, b + c).

Consider the basis for R3,

B 1 = b. c.R3 , B 2 =(0, 1, 0), (1, 0, 1), (1, 0, 0)

and consider the basis for R2,

B 1 = b. c.R2 , B 2 =(1, 1), (1, 0)

.

(a) Compute f (1, 2, 3) using the formula of the linear transformation.

(b) Find M (f ; B 1, B 1) and compute f (1, 2, 3) using this matrix.

(c) Find M (f ; B 2, B 1) and compute f (1, 2, 3) using this matrix.

(d) Find M (f ; B 1, B 2) and compute f (1, 2, 3) using this matrix.

(e) Find M (f ; B 2, B 2) and compute f (1, 2, 3) using this matrix.

5.43 Consider the basis for R3

B 1 =(1, −1, 0), (−1, 1, −1), (0, 1, 0)

, B 2 = b. c.R3

and

B 3 =(1, 1, 1), (0, 1, 1), (0, 0, 1)

.

Determine the transition matrix from:

(a) B 1 to B 2.

(b) B 2 to B 1.

(c) B 1 to B 3.

5.46 Let f : R3 −→ R2 the linear transformation such that

M (f ; b. c.R3 , b. c.R2) =

1 1 0

0 1 1

.

Consider the basis

B =(0, 1, 0), (1, 0, 1), (1, 0, 0)

and B =

(1, 1), (1, 0)

for R3 and for R2, respectively. Using transition matrices, compute:

26



(a) M (f ; B , b. c.R2).

(b) M (f ; b. c.R3 , B ).

(c) M (f ; B , B ).

5.110 Let E be a real vector space and let B = (u1, u2, u3) a basis for E . Consider the linear

transformation f : R5 −→ E such that, for all a,b, c, d, e ∈ R, is

f (a,b,c,d,e) = (−b − c + d)u1 + (2a + b + 3c − 3d)u2 + (b + c − d)u3.

.

(a) Determine M(f ; b. c.R5 , B ).

(b) Determine a basis for the range of f , Im f .

(c) In R5, consider the vectors

v1 = (2, 2, 0, 2, 2), v2 = (−1, −1, 1, 0, 1) and v3 = (0, 0, 0, 0, 1).

Show that (v1, v2, v3) is a basis for Ker f .

(d) Determine a basis for R5 that contains v1, v2 and v3.

(e) Consider B the basis that you have obtained in (d), determine M(f ; B , B

).

27





Chapter 6

Eigenvalues and Eigenvectors

6.1 Let f : R3 −→ R3 be the linear transformation such that, for all (a,b,c) ∈ R3 is

f (a,b,c) = (a + b,b, 2c).

Consider the vectors u1 = (2, 0, 0), u2 = (0, 0, 7) and u3 = (0, 0, 0). Indicate if each

of the vectors u1, u2, u3 is an eigenvector of f and, if this is true, which is the corre-

sponding eigenvalue.

6.2 Let A =

1 0

1 2

∈ M2×2(R).

(a) Show that

1

−1

and

0

2

are eigenvectors for A and indicate the corresponding

eigenvalues.

(b) Show that

0

α

and

α

−α

are eigenvectors for A and indicate the corresponding

eigenvalues.

6.3 Justify that if α is an eigenvalue of a matrix A ∈ Mn×n(C) then α is an eigenvalue of

A.

6.7 Let A ∈ Mn×n(K) be a matrix idempotent (that is, A2 = A).

(a) Show that if α is an eigenvalue of A then α ∈ {0, 1}.

(b) Indicate a matrix which has all eigenvalues in the set {0, 1} but that is not an

idempotent matrix.

6.12 Determine eigenvalues of the matrix

A =

2 −i 0

i 2 0

0 0 3

∈ M3×3(C)

and compute its respectively algebraic multiplicity.

29



6.13 Let A =

0 1

−1 0

, B =

−2 −1

5 2

∈ M2×2(K). Show that:

(a) If K

=R

then A has not eigenvalues.

(b) If K = C then A has two distinct eigenvalues.

(c) The matrices A and B have the same characteristic polynomial.

6.14 Let A =

0 2 0

−2 0 0

0 0 3

∈ M3×3(R).

(a) Compute the eigenvalues of A and its algebraic multiplicities.

(b) Compute the determinant of A using its eigenvalues.

6.19 Let A ∈ Mn×n(K) be an invertible matrix. Show that:

(a) If α is an eigenvalue of A then α = 0 and α−1 is an eigenvalue of A−1.

(b) If X ∈ Mn×1(K) is an eigenvector of A corresponding to the eigenvalue α, then

X is an eigenvector of A−1 corresponding to the eigenvalue α−1.

6.28 Let f : R3 −→ R3 be the linear transformation given by the formula

f (a,b,c) = (−b − c, −2a + b − c, 4a + 2b + 4c),

for all (a,b,c) ∈ R3. Find the eigenvalues and the corresponding eigenspace of f .

6.35 Consider the triangular matrices

A = −2 1

0 2

, B =

5 0

4 1

∈ M2×2(R).

Without computing eigenvalues, justify that A and B are both diagonalizable matrices

and indicate a diagonal matrix DA similar to A and a diagonal matrix DB similar to

B.

6.36 Let A =

2 5 2

0 3 0

2 −1 2

∈ M3×3(R). Without computing eigenspaces de A justify that

A is diagonalizable.

6.37 Consider the matrix

A =

3 2 0

−4 −3 0

4 2 −1

∈ M3×3(R).

(a) Compute the eigenvalues of A and find their algebraic multiplicities.

(b) Find a basis for each of the eigenspaces of A.

30



(c) Show that A is diagonalizable and indicate an invertible matrix P ∈ M3×3(R)

and a diagonal matrix D such that

P −1AP = D.

6.41 Let f be an endomorphism of R3 and B = (e1, e2, e3) a basis for R3. If

M (f ; B , B ) =

3 2 0

−4 −3 0

4 2 −1

,

Compute:

(a) The eigenvalues of f .

(b) A basis, B , for R3, whose elements are eigenvectors of f .

(c) M (f ; B , B ) such the basis B is the basis computed in (b).

Observation: Compare the results obtained in this exercise with the results that you

have obtained in the exercise 6.37.


A =

1 0 −1

1 2 1

2 2 3

, B =

0 1 0

0 0 1

1 −3 3

∈ M3×3(R).

(a) Compute the eigenvalues and its algebraic multiplicities for each of the previous

matrices.

(b) i. Show that A is diagonalizable.

ii. Find if B is diagonalizable.

(c) Find an invertible matrix P ∈ M3×3(R) such that P −1AP is a diagonal matrix

and the diagonal elements of P −1AP are in increasing order.

6.93 Let A ∈ M3×3(R) such that

A

1

2

3

=

2

4

6

, A

0

1

2

=

0

0

0

and A

0

0

1

=

0

0

2

.

(a) Compute the eigenvalues of A and its geometric multiplicities.

(b) Indicate, if there is a, diagonal matrix similar to A.

(c) Find a matrix A that verify the previous conditions.


F =

(x,y,z) ∈ R3 : x + 2y + z = 0

.

31



Let f : R3 −→ R3 the linear transformation such that (1, −1, 0) is an eigenvector of f

corresponding to the eigenvalue 2 and such that

f (a,b,c) = (0, 0, 0),

for all (a,b,c) ∈ F .

(a) Justify that B =

(1, −1, 0), (1, 1, −3), (1, 0, −1)

is a basis for R3 that has only

eigenvectors of f .

(b) Show that 0 is an eigenvalue of f and mg(0) = ma(0).

(c) Determine M(f ; B , b. c.R3).

Date post:	04-Jun-2018
Category:	Documents
Upload:	ceciliaperdigao
View:	219 times
Download:	0 times

Linear Algebra Exercises

Documents