This document provides examples and exercises for determining whether sets of vectors span vector spaces, are linearly independent, or can be expressed as linear combinations of other vectors. It includes problems involving vector spaces of matrices, real vectors, and polynomials. The tutorial aims to help students practice fundamental concepts in linear algebra through computational problems.

1. DEPARTMENT OF MATHEMATICS
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 0 - PREREQUISITE EXERCISE
1. Determine whether the following matrices are in Reduced Row Echelon form, Row
Echelon Form, or not in both forms.
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 0 0 1 0 0 0 1 0
⎢ ⎥
(a) ⎣ 0 0 0 ⎦ (b) ⎢ 0 0 1 ⎥
⎣ ⎦ (c) ⎢ 1 0 0 ⎥
⎣ ⎦
0 0 1 0 0 0 0 0 0
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 1 0 1 0 0 1 3 4
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(d) ⎣ 0 1 0 ⎦ (e) ⎣ 0 1 0 ⎦ (f) ⎣ 0 0 1 ⎦
0 0 0 0 2 0 0 0 0
⎡ ⎤
⎡ ⎤ 1 3 0 2 0
1 5 −3 ⎢ ⎥
1 0 2 2 0
(g) ⎢ 0 1
⎣ 1 ⎥⎦ (h) ⎢
⎢
⎥
⎥
⎣ 0 0 0 0 1 ⎦
0 0 0
0 0 0 0 0
2. Solve the following System of Linear Equations by using Gaussian Elimination
Method.
x + y + 2z = 8
(a) −x − 2y + 3z = 1
3x − 7y + 4z = 10
x − y + 2z − w = −1
2x + y − 2z − 2w = −2
(b)
−x + 2y − 4z + w = 1
3x − 3w = −3
3. Solve the system of linear equations in Question (2) by using Gauss-Jordan Elimi-
nation Method.
4. By using Elementary Row Operations on the Augmented Matrix [A|I], find the
inverse of the following matrix A.
⎡ ⎤ ⎡ ⎤
⎡ ⎤ 1 0 0 0 0 0 2 0
1/5 1/5 −2/5 ⎢ ⎥ ⎢ ⎥
1 3 0 0 1 0 0 1
(a) ⎢ 1/5 1/5 1/10 ⎥
⎣ ⎦ (b) ⎢
⎢
⎥
⎥ (c) ⎢
⎢
⎥
⎥
⎣ 1 3 5 0 ⎦ ⎣ 0 −1 3 0 ⎦
1/5 −4/5 1/10
1 3 5 7 2 1 5 −3
1

2. 5. Let
⎡ ⎤ ⎡ ⎤
1 2 3 1 0 5
⎢ ⎥ ⎢ ⎥
A = ⎣ 1 4 1 ⎦; B = ⎣ 0 2 −2 ⎦ .
2 1 9 1 1 4
Show that A and B are Row Equivalent. Find a sequence of elementary row opera-
tons that generates B from A.
6. Find the condition of bi (1 ≤ i ≤ 3) such that the following systems are Consistent
(in which the solution exists).
(a) x − 2y + 5z = b1 (b) x − 2y − z = b1
4x − 5y + 8z = b2 −4x + 5y + 2z = b2
−3x + 3y − 3z = b3 −4x + 7y + 4z = b3
@LWJ Semester I 2006/07
2

3. DEPARTMENT OF MATHEMATICS
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 1
1. Determine whether the given set V with the given operations is a vector space
or not. For those that are NOT, list all axioms that fail to hold.
(a) V is the set of all 2 × 2 non-singular matrices.
The operations of addition and scalar multiplication are the standard matrix
operations.
(b) V = { v = (v1 , v2 , v3 ) ∈ R3 | v1 = v3 }.
¯
The operations of addition and scalar multiplication are the standard oper-
ations on R3 .
(c) V = { v = (v1 , v2 , v3 ) ∈ R3 | v1 = v2 }.
¯
The operations of addition and scalar multiplication are the standard oper-
ations on R3 .
x1
2. Let V = x=
¯ x1 , x2 ∈ R . For u and v in V and k ∈ R, define
¯ ¯
x2
addition and scalar multiplication operations as follows:
u1 v1 u1 + v1
u+v =
¯ ¯ + =
u2 v2 u2 + v2
u1 ku1 + 1
k¯ = k
u = .
u2 ku2 − 1
(a) If u = (−1, 2) and v = (3, −4), compute (i) u + v ; (ii) 1 u; (iii) 5¯ + 5¯;
¯ ¯ ¯ ¯ 3
¯ u v
(iv) 5(¯ + v ); (v) (2 + 3)¯; (vi) −2¯ + 3¯.
u ¯ u u u
(b) Find an object 0 ∈ V such that ¯ + u = u for any u ∈ V .
¯ 0 ¯ ¯ ¯
(c) If the object 0 in (b) exists, find an object w ∈ V such that u + w = ¯
¯ ¯ ¯ ¯ 0.
(This object w is called the negative of u and is denoted by −¯.)
¯ ¯ u
(d) Is 1¯ = v for each v ∈ V ?
v ¯ ¯
(e) Is V with the given two operations a vector space?
1

4. x1
3. Let V = x=
¯ x1 , x2 ∈ R . For u and v in V and k ∈ R, define
¯ ¯
x2
addition and scalar multiplication operations as follows:
u1 v1 u1 v1
u+v =
¯ ¯ + =
u2 v2 u2 + v2
u1 u1 + k
k¯ = k
u = .
u2 ku2
(a) If u = (−2, 7) and v = (1, −2), compute (i) u + v ; (ii) 1 u; (iii) 3¯ + 3¯;
¯ ¯ ¯ ¯ 2
¯ u v
(iv) 3(¯ + v ); (v) (2 + 5)¯; (vi) 2¯ + 5¯.
u ¯ u u u
(b) Find an object ¯ ∈ V such that ¯ + u = u for any u ∈ V .
0 0 ¯ ¯ ¯
(c) If the object ¯ in (b) exists, find an object w ∈ V such that u + w = ¯
0 ¯ ¯ ¯ 0.
(This object w is called the negative of u and is denoted by −¯.)
¯ ¯ u
(d) Is (k + l)¯ = k¯ + l¯ for each v ∈ V and k, l ∈ R?
v v v ¯
(e) Is V with the given two operations a vector space?
4. Let V = R2 . For u and v in V and k ∈ R, we define addition and scalar
¯ ¯
multiplication operations as follows:
u + v = (u1 , u2 ) + (v1 , v2 ) = (u1 + v1 , 0)
¯ ¯
k¯ = k(u1 , u2 ) = (ku1 , ku2 ).
u
(a) If u = (−3, 2) and v = (−1, 5), compute (i) u + v ; (ii) 1 u; (iii) 2¯ + 2¯;
¯ ¯ ¯ ¯ 2
¯ u v
(iv) 2(¯ + v ); (v) (2 + 5)¯; (vi) 2¯ + 5¯.
u ¯ u u u
(b) Find an object ¯ ∈ V such that ¯ + u = u for any u ∈ V .
0 0 ¯ ¯ ¯
(c) If the object ¯ in (b) exists, find an object w ∈ V such that u + w = ¯
0 ¯ ¯ ¯ 0.
(This object w is called the negative of u and is denoted by −¯.)
¯ ¯ u
(d) Is k(¯ + w) = k¯ + k w for each v , w ∈ V and k ∈ R?
v ¯ v ¯ ¯ ¯
(e) Is V with the given two operations a vector space?
5. Determine whether the given set W is a subspace of the vector space V .
(a) W = { (x1 , x2 , x3 ) ∈ R3 | x1 = x2 − x3 }, and V = R3 .
(b) W = { (x, y) ∈ R2 | x ≥ 0 }, and V = R2 .
(c) W = { (x, y) ∈ R2 | y = 2x }, and V = R2 .
(d) W = { p(x) ∈ P2 | p(0) = 0 }, and V = P2 .
(e) W = { ax2 + 3x − 2 | a ∈ R }, and V = P2 .
a c
(f) W = a, c ∈ R , and V = M22 .
c −a
a b
(g) W = a, b, c ∈ R , and V = M22 .
c 0
@LWJ Semester I 2006/07
2

5. DEPARTMENT OF MATHEMATICS
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 2
1. Determine whether the following vector is a linear combination of u = (0, −2, 2) and
¯
v = (1, 3, −1) or not.
¯
(a) w = (2, 2, 2)
¯ (c) w = (0, 0, 0)
¯
(b) w = (3, 1, 5)
¯ (d) w = (0, 4, 5)
¯
2. Determine whether the following vectors span R3 or not.
(a) u = (3, 2, −1), v = (1, 2, −3).
¯ ¯
(b) u = (1, 0, 3), v = (3, 1, 0), w = (1, 2, 3).
¯ ¯ ¯
(c) v1 = (2, 1, −2), v2 = (6, 3, −6), v3 = (−2, −1, 2).
¯ ¯ ¯
(d) u = (3, 2, −2), v = (1, 1, 0), w = (−2, 1, 2).
¯ ¯ ¯
(e) v1 = (2, 4, −3), v2 = (1, 2, 1), v3 = (−2, 1, 5), v4 = (0, 4, 1).
¯ ¯ ¯ ¯
3. Determine whether the following vectors span the vector space V or not.
0 0 1 1 −1 1 0 0
(a) u1 =
¯ , u2 =
¯ , u3 =
¯ , u4 =
¯ ; V = M22 .
1 0 0 0 1 0 0 1
1 0 0 2 0 0 0 0
(b) w1 =
¯ , w2 =
¯ , w3 =
¯ , w4 =
¯ ; V = M22 .
0 0 0 0 3 0 0 4
(c) p1 = 1, p2 = x + 1, p3 = x2 + 1; V = P2 .
¯ ¯ ¯
(d) q1 = 2, q2 = x + 1, q3 = x2 + x − 1; V = P2 .
¯ ¯ ¯
4. Determine whether the following vectors are linearly independent or linearly dependent
in the vector space V .
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
0 2 1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
(a) u = ⎣ −1 ⎦, v = ⎣ 0 ⎦, w = ⎣ 1 ⎦;
¯ ¯ ¯ V = R3 .
3 −3 0
0 0 1 0
(b) v1 =
¯ , v2 =
¯ ; V = M22 .
1 −1 −1 0
(c) p = 3x2 − x − 1, q = x + 1; V = P2 .
¯ ¯
(d) x1 = (0, −3, 1, 1), x2 = (5, 5, 1, 3), x3 = (−1, 0, 5, 1); V = R4 .
¯ ¯ ¯
1

6. 5. Let v1 = (−1, 3), v2 = (2, 1) ∈ R2 . Are w = (−1, 2) and w2 = (−3, 5) elements in
¯ ¯ ¯ ¯
span({¯1 , v2 })?
v ¯
6. Find the value of α such that the following vectors are linearly dependent in R2 .
1
α − 12
(a) x1 =
¯ , x2 =
¯
−3 α
α 2
(b) y1 =
¯ , y2 =
¯
2 α
7. Find the value of λ such that the following vectors are linearly dependent in R3 .
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
λ −1 −1
x1 = ⎢ −1 ⎥ ,
¯ ⎣ ⎦ x2 = ⎢ λ ⎥ ,
¯ ⎣ ⎦ x3 = ⎢ −1 ⎥ ,
¯ ⎣ ⎦
−1 −1 λ
8. Prove that if { v1 , v2 } is linearly independent and v3 ∈ span ({ v1 , v2 }), then { v1 , v2 , v3 }
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
is also linearly independent.
9. Prove Theorem 1.1.3 in Chapter 1.
@LWJ Semester I 2006/07
2

7. DEPARTMENT OF MATHEMATICS
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 3
1. Let v1 = (2, 1), v2 = (5, 3), v3 = (−4, 7).
¯ ¯ ¯
(a) Is { v1 , v2 } a basis for R2 ?
¯ ¯
(b) Is { v1 , v2 , v3 } a basis for R2 ?
¯ ¯ ¯
(c) Is span ({ v1 , v2 })= span ({ v1 , v2 , v3 })?
¯ ¯ ¯ ¯ ¯
2. Explain why the following sets are linearly dependent.
(a) u = (4, −1, 2), v = (2, −1/2, 1).
¯ ¯
(b) u1 = (1, 2, 3), u2 = (0, 0, 0), u3 = (3, 2, 1).
¯ ¯ ¯
(c) p1 = 2x2 − 4, p2 = x2 − 2.
¯ ¯
2 −1 4 −2
(d) A1 = , A2 = .
1 3 2 6
(e) v1 = (1, 2, 3), v2 = (2, 1, 3), v3 = (3, 1, 2), v4 = (2, 3, 1).
¯ ¯ ¯ ¯
3. Let S = { v1 , v2 , v3 } with
¯ ¯ ¯
v1 = (3, 1, −4),
¯ v2 = (2, 5, 6),
¯ and v3 = (1, 4, 8).
¯
(a) Show that S is a basis for R3 .
(b) Find the coordinate vector of w = (5, 2, −1) relative to the basis S.
¯
(c) Find a vector w in R3 with (w)S = (1, −1, 2) is the coordinate vector
¯ ¯
relative to S.
4. Show that
p1 = x2 ,
¯ p2 = x2 + x,
¯ p3 = x2 + x + 1
¯
form a basis for P2 .
5. Show that the following set of vectors is a basis for M22 .
0 0 1 1 −1 1 0 0
, , , and .
1 0 0 0 1 0 0 1
1

8. 6. Determine whether the following sets of vectors are bases for M22 or not.
0 1 1 0 1 0 0 0
, , , and .
2 0 −1 0 −1 1 −1 3
7. Determine whether the following sets of vectors are bases for P2 or not.
p1 = x2 + 1,
¯ p2 = x + 1,
¯ p3 = x2 − x + 2.
¯
8. If x1 , . . . , xm form a basis for a subspace S, then prove that
¯ ¯
x1 , x1 + x2 , . . . , x1 + x2 + · · · + xm
¯ ¯ ¯ ¯ ¯ ¯
also form a basis for S.
Semester I 2006/07
2

9. DEPARTMENT OF MATHEMATICS
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 4
1. Let u = (u1 , u2 , u3 ) and w = (w1 , w2 , w3 ). Determine whether the following
¯ ¯
definitions are inner products on R3 or not. If not, list all axioms that are
not satisfied.
(a) < u, w >= u2 w2 + u3 w3 .
¯ ¯
(b) < u, w >= u1 w1 + 2u2 w2 + 3u3 w3 .
¯ ¯
(c) < u, w >= u1 w1 + u2 w2 − u3 w3 .
¯ ¯
2. Find w with w = (2, −5) if the inner product on R2 is
¯ ¯
(a) the Euclidean inner product.
(b) the weighted inner product < u, v >= 5u1 v1 + 2u2 v2 with u = (u1 , u2 )
¯ ¯ ¯
and v = (v1 , v2 ).
¯
(c) the inner product generated by matrix
−2 3
A= .
2 7
3. Use the inner products defined in Question 2 to find d(¯, w) if u = (−1, 3)
u ¯ ¯
and w = (3, 5).
¯
4. Let M22 have the inner product:
u1 u2 v1 v2
, = u1 v1 + u2 v2 + u3 v3 + u4 v4 .
u3 u4 v3 v4
Find A and d(A, B) if
2 4 −4 2
(a) A = dan B=
−3 1 5 1
6 −1 −1 8
(b) A = dan B=
7 4 0 2
1

10. 5. Let P2 have the inner product:
1
p, q =
¯ ¯ p(x)q(x)dx.
0
Find p and d(¯, q ) if
¯ p ¯
(a) p = 3x2 − 2 and q = x2 + x.
¯ ¯
(b) p = x2 + x + 1 and q = 5x2 − x + 3.
¯ ¯
6. Let u, v and w be vectors such that
¯ ¯ ¯
< u, v >= 3, < v , w >= −2, < u, w >= 7, u = 2, v = 5, w = 8.
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
Calculate the following:
(a) < 2¯ + v , v − w >.
u ¯ ¯ ¯
(b) < u − v + 2w, v + 3w >.
¯ ¯ ¯ ¯ ¯
(c) 2¯ + w .
u ¯
(d) u − 3¯ + w .
¯ v ¯
7. Prove Theorem 1.5.1 (c) and (d) in Chapter 1.
8. Verify the Cauchy-Schwarz inequality for the following vectors.
(a) u = (2, 3) and v = (0, −1) are vectors in R2 with the inner product as
¯ ¯
in Question 2(b).
2 6 −3 1
(b) U = and V = are vectors in M22 with the
1 −3 4 2
inner product as in Question 4.
(c) p = 2x2 − x + 1 and q = x2 − 4 are vectors in P2 with the inner product
¯ ¯
as in Question 5.
9. Prove Theorem 1.5.3 (b), (c), (f) and (h) in Chapter 1.
Semester I 2006/07
2

11. DEPARTMENT OF MATHEMATICS
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 5
1. Find the nullspace of the following matrices.
3 4 −2 6
(a) A = (c) A =
−1 2 1 −3
⎡ ⎤ ⎡ ⎤
1 0 3 1 5 1
(b) A = ⎣ −1 1 −1 ⎥
⎢
⎦
⎢
(d) A = ⎣ 0 2 −3 ⎥
⎦
2 4 0 6 −2 4
2. Find a basis and dimension for the solution space of the following homoge-
neous systems.
(a) x1 − 2x2 + 3x3 = 0 (b) x1 − 3x2 + x3 + x4 = 0
2x1 + x2 − x3 = 0 −x1 − 5x2 + x3 − x4 = 0
x1 − 3x2 + 4x3 = 0 3x1 + x2 − 2x3 − 2x4 = 0
3. For the matrix A and ¯ as below, determine whether ¯ is in the clolumn
b b
space of A, if so, express ¯ as a linear combination of the column vectors of
b
A.
−5 2 ¯= 8
(a) A = ; b .
3 7 −1
⎡ ⎤ ⎡ ⎤
7 2 1 2
⎢ ⎥ ¯ = ⎢ 8 ⎥.
(b) A = ⎣ 2 3 4 ⎦ ; b ⎣ ⎦
1 1 0 −1
4. Find a basis for the nullspace of the following A, if exists.
⎡ ⎤ ⎡ ⎤
3 −1 0 1 2 −1 −2
(a) A = ⎢ 1
⎣ 7 0 ⎥
⎦ (c) A = ⎢ −3 3
⎣ 4 1 ⎥
⎦
2 4 0 5 1 1 2
⎡ ⎤ ⎡ ⎤
−1 3 5 7 1 6
⎢ ⎥ ⎢
(b) A = ⎣ −6 4 1 ⎦ (d) A = ⎣ 1 4 5 ⎥
⎦
−3 −2 1 2 2 −1
5. Find the bases for row space and column space of the matrix A in question
no. 4.
6. Find a basis for the row space of the matrix A in question no. 4 consisting
entirely of row vectors of A.
1

12. 7. Find a basis for the space spanned by the vectors:
(a) (3, 0, 3, −2), (3, −4, −1, −1), (5, −4, 1, 1).
(b) (−2, 1, 3, 4), (3, −1, 3, 2), (−1, −5, 2, −3).
(c) (0, 1, 0, 1), (1, 1, 0, 0), (−1, 1, 0, 1).
8. Find a subset of S = { v1 , v2 , v3 , v4 } that forms a basis for the space spanned
¯ ¯ ¯ ¯
by the vectors in S; then express each vector not in the basis as a linear
combination of the basis vectors.
(a) v1 = (−3, 5, 1, −5), v2 = (−3, 3, −1, −7), v3 = (3, −1, 3, 9), v4 =
¯ ¯ ¯ ¯
(0, 1, 1, 1).
(b) v1 = (1, 1, 0, 1), v2 = (−1, 3, 2, 6), v3 = (−1, 5, 3, −2), v4 = (3, 0, 5, −1).
¯ ¯ ¯ ¯
9. Find the dimension of
(a) row space of A.
(b) column space of A.
(c) nullspace of A.
(d) nullspace of AT .
if given that
(i) A is a 4 × 4 matrix and rank (A) = 2.
(ii) A is a 7 × 5 matrix and rank (A) = 3.
(iii) A is a 3 × 3 matrix and rank (A) = 0.
10. Prove that if A is a 5 × 6 matrix, then the column vectors in A are linearly
dependent.
11. Prove that rank (7A) = rank (A).
12. Show that if A is a square matrix, then nullity (A) is equal to nullity (AT ).
@LWJ Semester I 2006/07
2

13. JABATAN MATEMATIK
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 6
1. Let M22 be an inner product space as in Question 4 in Tutorial 4. Determine
whether the following pairs of vectors are orthogonal or not.
1 0 3 6
(a) A = and B =
0 1 5 −3
1 2 4 3
(b) A = and B =
3 4 2 1
2. Let R4 be an Euclidean space and u = (1, −1, 2, 0). Determine whether the
¯
vector u is orthogonal to the set of vectors W = { w1 , w2 , w3 } where
¯ ¯ ¯ ¯
(a) w1 = (0, 0, 0, 3), w2 = (−3, 3, 3, −8), w3 = (2, −2, 3, 1).
¯ ¯ ¯
(b) w1 = (3, −3, −3, 7), w2 = (0, 2, 1, 5), w3 = (6, 0, −3, 6).
¯ ¯ ¯
3. Find a basis for the orthogonal complement of the subspace of R3 spanned
by the vectors.
(a) v1 = (3, 1, 1), v2 = (−4, 4, 5), v3 = (2, 6, 7).
¯ ¯ ¯
(b) v1 = (1, 2, −3), v2 = (3, 6, −9).
¯ ¯
4. Determine whether the following sets of vectors are orthogonal on R3 with
respect to the Euclidean inner product.
√ √ √ √ √
(a) u = (−1/ 2, 0, 1/ 2), v = (1/ 6, −2/ 6, 1/ 6).
¯ ¯
√ √
(b) u = (0, 1, 0), v = (0, 0, 1), w = (1/ 2, 1/ 2, 0).
¯ ¯ ¯
5. Determine whether the sets of vectors in question no. 5 are orthonormal on
R3 with respect to the Euclidean inner product.
6. Verify that the following sets of vectors are orthogonal sets with respect to
euclidean inner product; then transform these sets to orthonormal sets.
(a) u1 = (2, 0, 3), u2 = (0, 6, 0), u3 = (−3, 0, 2).
¯ ¯ ¯
(b) u1 = (−1/5, −1/5, −1/5), u2 = (−1/3, −1/3, 2/3), u3 = (1/2, −1/2, 0).
¯ ¯ ¯

14. 7. Show that S = { u1 , u2 , u3 } with
¯ ¯ ¯
3 4 4 3
u1 = (0, 0, 1),
¯ u2 = (− , , 0),
¯ and u3 = ( , , 0)
¯
5 5 5 5
is an orthonormal basis for the Euclidean space R3 . Express w = (2, 1, −3)
¯
as a linear combination of vectors in S and obtain the coordinate vector
(w)S .
¯
8. Show that S = { v1 , v2 , v3 , v4 } with
¯ ¯ ¯ ¯
v1 = (1, −2, 3, −4),
¯ v2 = (2, 1, −4, −3),
¯ v3 = (−3, 4, 1, −2),
¯ and v4 = (4, 3, 2, 1))
¯
is an orthogonal basis for the Euclidea space R4 . Then express each of the
following vectors as a linear combination of vectors in S.
(a) w = (2, 1, 1, 2)
¯
(b) w = (5, 5, −2, −2)
¯
9. Let R3 be the Euclidean space. Use the Gram-Schmidt process to transform
the following basis vectors into an orthonormal basis.
(a) u1 = (1, 1, −1), u2 = (−1, 1, 0), u3 = (1, −2, 1).
¯ ¯ ¯
(b) u1 = (1, 0, 0), u2 = (4, 5, −2), u3 = (0, 3, 1).
¯ ¯ ¯
10. Let R3 have the inner product < u, v >= u1 v1 + 2u2 v2 + 3u3 v3 . Use
¯ ¯
the Gram-Schmidt process to transform u1 = (1, 1, 1), u2 = (1, −1, 0),
¯ ¯
u3 = (1, 0, 0) into an orthonormal basis.
¯
@LWJ Semester I 2006/07

15. JABATAN MATEMATIK
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 7
1. Show that ⎡ ⎤
3/7 2/7 6/7
A = ⎢ −6/7 3/7 2/7 ⎥
⎣ ⎦
2/7 6/7 −3/7
is an orthogonal matrix by
(a) computing AAT .
(b) using part (b) of Theorem 3.3.1.
(c) using part (c) of Theorem 3.3.1.
2. Find the inverse of the above matrix A.
3. Let B = { u1 , u2 } and B = { u1 , u2 } where
¯ ¯ ¯ ¯
1 2 2 3
u1 =
¯ , u2 =
¯ ; u1 =
¯ , u2 =
¯
3 2 1 −4
be two bases for R2 .
(a) Find the transition matrix from B to B .
(b) Find the transition matrix from B to B.
−1
(c) Use the equation [¯]B = P [¯]B to find [w]B if [w]B =
v v ¯ ¯ .
3
4. Let B = { u1 , u2 , u3 } and B = { u1 , u2 , u3 } where
¯ ¯ ¯ ¯ ¯ ¯
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
1 −1 −1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
u1 = ⎣ 2 ⎦ ,
¯ u2 = ⎣ −1 ⎦ ,
¯ u3 = ⎣ 1 ⎦ ;
¯
1 1 7
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
3 0 1
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
u1 = ⎣ 7 ⎦ ,
¯ u2 = ⎣ 4 ⎦ ,
¯ u3 = ⎣ 0 ⎦
¯
−2 1 0
be two bases for R3 .
1

16. (a) Find the transition matrix from B to B .
(b) Find the transition matrix from B to B.
⎡ ⎤
−2
(c) Compute the coordinate matrix [w]B where w = ⎣ −6 ⎥ and use the
¯ ¯ ⎢
⎦
4
equation [¯]B = P −1 [¯]B to calculate [w]B .
v v ¯
(d) Check your answer by computing [w]B directly.
¯
5. Let B = { p1 , p2 } and B = { p1 , p2 } where
¯ ¯ ¯ ¯
p1 = 3 + x,
¯ p2 = −4 + 2x;
¯ p1 = 2,
¯ p2 = 1 + 3x
¯
be two bases for P1 .
(a) Find the transition matrix from B to B .
(b) Compute the coordinate matrix [¯]B where q = −1 + 4x, and use the
q ¯
equation [¯]B = P [¯]B to calculate [¯]B .
v v q
(c) Check your answer by computing [¯]B directly.
q
@LWJ Semester I 2006/07
2

17. JABATAN MATEMATIK
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 8
1. Find bases for the eigen spaces for each of the following matrices.
−1 3 2 −6
(a) (d)
8 4 4 −8
⎡ ⎤ ⎡ ⎤
1 0 0 2 0 0
⎢ ⎥ ⎢ ⎥
(b) ⎣ −8 4 −6 ⎦ (e) ⎣ 1/2 3 1 ⎦
8 1 9 0 0 2
⎡ ⎤ ⎡ ⎤
7 10 6 −7 −9 3
⎢ ⎥ ⎢
(c) ⎣ 2 −1 −6 ⎦ (f) ⎣ 2 4 −2 ⎥
⎦
−2 −5 0 −3 −3 −1
2. Find the eigen values for eigen spaces of A11 if
⎡ ⎤ ⎡ ⎤
1 −1 −1 2 0 0
⎢
(a) A = ⎣ 1 2 −2 ⎥
⎦ (b) A = ⎣ 1 −1 −2 ⎥
⎢
⎦
0 −1 0 −1 0 1
3. By using Theorem 4.1.7, verify that the following matrices are invertible.
⎡ ⎤ ⎡ ⎤
3 −1 1 5 2 6
⎢
(a) ⎣ 1 0 1 ⎥
⎦ (b) ⎣ 0 −8 −1 ⎥
⎢
⎦
−5 2 −3 1 −2 0
4. Verify Theorem 4.1.8 for the following matrices.
⎡ ⎤ ⎡ ⎤
1 0 2 −1 1 0
(a) ⎣ −3 0 −2 ⎥
⎢
⎦ (c) ⎣ 1 −2 1 ⎥
⎢
⎦
0 1 5 0 −1 1
⎡ ⎤ ⎡ ⎤
0 3 −1 4 3 0
(b) ⎢ 2 −5
⎣ 5 ⎥
⎦ (d) ⎢ 2 3 0 ⎥
⎣ ⎦
0 −1 3 0 0 6
5. Find a matrix P (if any) that diagonalizes each of the following A, and
determine P −1 AP .
1

18. ⎡ ⎤ ⎡ ⎤
−9 −6 −22 3 1 1
⎢
(a) A = ⎣ 1 2 2 ⎥
⎦
⎢
(c) A = ⎣ 1 1 0 ⎥
⎦
4 2 10 −5 −3 −2
⎡ ⎤ ⎡ ⎤
2 0 0 1 0 0
(b) A = ⎢ −1
⎣ 2 0 ⎥
⎦ (d) A = ⎢ −8 4 −5 ⎥
⎣ ⎦
4 −3 5 8 0 9
4 −3
6. Let A = . Find a nonsingular matrix P such that P −1 AP =
1 0
1 0
and hence prove that
0 3
3n − 1 3 − 3n
An = A+ I2 .
2 2
7. Use Theorem 4.2.4 to find A10 if
⎡ ⎤ ⎡ ⎤
1 1 0 2 0 1
⎢ ⎥ ⎢ ⎥
(a) A = ⎣ −1 −2 −1 ⎦ (b) A = ⎣ 6 4 −3 ⎦.
3 1 −2 2 0 3
8. Find a matrix P that orthogonally diagonalizes A, and determine P −1 AP ,
if
1 −3 5 −6
(a) A = (d) A =
−3 9 −6 −11
⎡ ⎤ ⎡ ⎤
2 −1 −1 4 −1 2
⎢
(b) A = ⎣ −1 1 0 ⎥
⎦
⎢
(e) A = ⎣ −1 4 −2 ⎥
⎦
−1 0 1 2 −2 3
⎡ ⎤ ⎡ ⎤
1 1 0 0 0 0 0 0
⎢ 1 1 0 0 ⎥ ⎢ 0 1 0 0 ⎥
⎢ ⎥ ⎢ ⎥
(c) A = ⎢ ⎥ (f) A = ⎢ ⎥
⎣ 0 0 0 0 ⎦ ⎣ 0 0 4 6 ⎦
0 0 0 2 0 0 6 −1
@LWJ Semester I 2006/07
2

19. JABATAN MATEMATIK
UNIVERSITI PUTRA MALAYSIA
MTH 3201
TUTORIAL 9
1. Check whether the function T defined by the following formula is a linear
transformation or not.
(a) T : R3 → R2 ; T ((x, y, z)) = (x − y, y − z)
(b) T : P2 → P2 ; T (ax2 + bx + c) = a(x + 1)2 + b(x + 1) + c
a b
(c) T : M22 → R; T =a+b−c−d
c d
x y
(d) T : R2 → R; T ((x, y)) =
x+y x−y
2. Suppose that the linear operators T1 : P3 → P3 and T2 : P3 → P3 are, respec-
tively defined by the formulas
T1 (p(x)) = p(x − 1) and T2 (p(x)) = p(x + 2).
Find (a) (T2 ◦ T1 )(p(x)) and (b) (T1 ◦ T2 )(p(x)).
3. Let T1 : M22 → R and T2 : M22 → M22 be the linear transformations de-
fined, respectively by
a b a b a c
T1 = a − b + 4c − d dan T2 = .
c d c d b d
Find (if no, explain your answer.)
a b
(a) (T1 ◦ T2 )
c d
a b
(b) (T2 ◦ T1 )
c d
1

20. 4. Let T : P2 → P4 be the linear transformation defined by
T (p(x)) = x2 p(x).
(a) Determine whether the vectors (i) x2 + x, (ii) x + 1 and (iii) 3 − x2 lie
in range (T ).
(b) Determine whether the vectors (i) x2 , (ii) 0 and (iii) x + 1 lie inberada
dalam kernel (T ).
5. Let T : R3 → R3 be multiplication by the following matrices A. Find a basis
for kernel (T ) and a basis for range (T ) if
⎡ ⎤ ⎡ ⎤
4 5 7 1 −1 3
⎢ ⎥ ⎢
(a) A = ⎣ −6 1 −1 ⎦ (b) A = ⎣ 5 6 −4 ⎥
⎦
3 6 4 7 4 2
6. Let T : R3 → R3 be a linear operator given by the formula
T ((x, y, z)) = (2x + 4y − 6z, x − 2y + z, 5x − 2y − 3z).
Find
(a) basis for kernel (T ) and
(b) basis for range (T ).
@LWJ Sem I 2006/07
2