linear algebra exercises
TRANSCRIPT
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 2/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 3/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 4/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 5/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 6/32
1.57 Consider the matrices
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 7/32
1.66 Consider the matrices
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 8/328
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 9/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 10/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 11/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 12/3212
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 13/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 14/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 15/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 16/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 17/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 18/32
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)
is linearly independent.
18
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 19/32
(c) The sequence
S = (u1 − u2, u2 − u3, u1 − u3)
is linearly independent.
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 20/32
(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.
4.69 Consider the subspace of R4,
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 21/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 22/3222
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 23/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 24/32
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.
(a) f : R3 −→ R2 such that
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,
for all (a,b,c) ∈ R3.
(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.
(a) f : R3 −→ R3 such that
f (a,b,c) = (2a, b + c, b − c),
for all (a,b,c) ∈ R3.
(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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 25/32
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.
(a) f : R3 −→ R3 such that
f (a,b,c) = (2a, b + c, b − c),
for all (a,b,c) ∈ R3.
(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.
(a) f : R4 −→ R4 such that
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 26/32
(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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 27/32
(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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 28/3228
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 29/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 30/32
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
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 31/32
(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.
6.89 Consider the matrices
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.
6.95 Consider the subspace of R3,
F =
(x,y,z) ∈ R3 : x + 2y + z = 0
.
31
8/13/2019 Linear Algebra Exercises
http://slidepdf.com/reader/full/linear-algebra-exercises 32/32
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).