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§3.2 Vector spaces
1. Give the definition of a vector space.
2. Let V be a vector space with vector addition ⊕ and scalar multiplication .
(a) Show that a zero vector of V is unique. (We denote the zero vector by 0).
(b) Let v ∈ V . Show that a negative of v is unique.
(c) Show that
0v =0
for all v ∈ V .
(d) Show that
c 0 = 0 for all c ∈ R
3. Let P be the set of all polynomials in t with coefficients in R.
(a) Let p(t) ∈ P . What is the degree of p(t)?
(b) We say that two polynomials p(t) and q(t) in P are equal
if
.
(c) Show that the set P with usual polynomial addition ⊕ and scalar multiplication is
a vector space.
4. Exercises §3.2 #6, 10, 20, 21.
§3.3 Subspaces
1. Let V be a vector space with operations ⊕ and , and let W be a nonempty subset of V .
(a) We say that W is closed under ⊕ and if
(b) W is called a subspace of V
if
.
.
This note is based on the textbook Elementary Linea Algebra written by Kolman and Hill.
1
(c) To determine whether W is a subspace of V or not, one only need to show that
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2. For a positive integer n, let Pn be the subset of P consisting of all polynomials of degree
less than or equal to n. Prove that Pn is a subspace of P .
3. Show that the subset

W =

a


3
 b  ∈ R | a, b ∈ R
a − 2b
of R3 is a subspace of R3 with usual vector addition and scalar multiplication.
4. Let V be a vector space, and let v1 , v2 , . . . , vk ∈ V .
(a) A vector v ∈ V is called a linear combination of v1 , v2 , . . . , vk
if
.
(b) What is Span S, where S = {v1 , v2 , . . . , vk }?
(c) Show that Span S is a subspace of V .
5. Let A be an m × n matrix.
(a) What is the solution space of the homogeneous system Ax = 0?
(b) Show that the solution space of Ax = 0 is a subspace of Rn .
6. Exercises § 3.3 #1, 2, 3, 9, 19, 29.
§3.4 Span and linear independence


v1
 
 v2 
n

1. Let A be an m × n matrix with columns a1 , a2 , . . . , an , and let v = 
 ..  ∈ R .
.
vn
Prove the relation
Av = v1 a1 + v2 a2 + · · · + vn an .
2. Let V be a vector space, and let v1 , v2 , . . . , vk ∈ V . We say that v1 , v2 , . . . , vk are
.
linearly independent if
3. (a) Consider the three vectors
v1 = [ 1 0 1 2 ],
v2 = [ 0 1 1 2 ],
v3 = [ 1 1 1 3 ]
in M14 . Determine whether they are linearly independent or not.
2
(b) Consider the vectors
w1 = t2 + t + 2,
w2 = 2t2 + t,
w3 = 3t2 + 2t + 2
in P2 . Determine whether they are linearly independent or not.
4. Let V be a vector space.
(a) let S1 and S2 be nonempty finite subsets of V such that S1 ⊆ S2 .
Show that if S2 is linearly independent, then so is S1 .
(b) Show that a finite subset S of V containing 0 is always linearly dependent.
5. Let V be a vector space, and let S = {v1 , v2 , . . . , vn } be a set of nonzero vectors in V
with n ≥ 2. Prove the assertion that
S is linearly dependent
⇐⇒
there is a vector vj with j ≥ 2 that can be written as a linear combination of its
preceding vectors v1 , v2 , . . . , vj−1 .
6. Exercises § 3.4 #5, 16, 17, 18, 21, 24.
§3.5 Basis and dimension
1. Let V be a vector space. A basis for V is defined to be a nonempty finite subset S of V
such that
.
2. Let n be a positive integer.
(a) For each 1 ≤ i ≤ n, let ei be the vector in Rn whose i-th entry is 1 and all other
entries are 0. Show that {e1 , e2 , . . . , en } is a basis for Rn .
(b) Show that {tn , tn−1 , . . . , t, 1} is a basis for Pn .
3. Let W be a subset of P2 given by
W = {at2 + bt + c | a, b, c ∈ R such that c = a − b}.
(a) Show that W is a subspace of P2 .
(b) Find a basis for W .
4. We say that a vector space V is finite-dimensional
if
.
5. Let V be a nonzero finite-dimensional vector space with a basis S = {v1 , v2 , . . . , vn }.
(a) Prove that every vector in V can be written in one and only one way as a linear
combination of the vectors in S.
3
(b) Let T = {w1 , w2 , . . . , wr } be a set of linearly independent vectors in V . What can
you say about r and n? (7
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(c) Let U = {u1 , u2 , . . . , um } be another basis for V . Prove that m = n.
6. (a) Give the definition of the dimension of a finite-dimensional vector space V . Explain
briefly why your definition makes sense.
(b) Explain the reason for
dim(Rn ) = n
and
dim(Pn ) = n + 1,
where n is a positive integer.
(c) Let V be a vector space of dimension n (> 0), and let T = {w1 , w2 , . . . , wm } be a
linearly independent set of vectors in V . Show that if m = n, then T is a basis for V .
(d) Consider a subset
S = {t2 + 1, t − 1, 2t + 2}
of P2 . Determine whether S is a basis for P2 or not. (Note that dim(P2 ) = 3. One
can use the preceding Problem 6(c), that is, if S is linearly independent, then it is a
basis for P2 .)
7. Exercises §3.5 #2, 3, 6, 11, 12, 13, 15, 21, 36, 41, 42, 48.
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§3.6 Homogeneous systems
1. Let A be an m × n matrix.
(a) What is the null space of A?
(b) What is the nullity of A?
(c) Give a formula for the nullity of A.
2. Exercises §3.6 #9, 24, 25.
§3.7 Coordinates
1. Let V be a vector space of dimension n (> 0) with an ordered basis S = {v1 , v2 , . . . , vn }.
Let v ∈ V . Give the definition of the coordinate vector of v with respect to the ordered
basis S.
2. Let S = {v1 , v2 , v3 } be an ordered subset of R3 , where
 
 
 
1
2
0
 
 
 
v1 = 1 , v2 = 0 , v3 = 1 .
0
1
2
4
(a) Show that S is a basis for R3 . (One can use the above Problem §3.5, 6(c).)


1


(b) Let v =  1 . Compute [v]S .
−5
3. Let V be a vector space of dimension n (> 0). Let
S = {v1 , v2 , . . . , vn }
and T = {w1 , w2 , . . . , wn }
be two ordered bases for V . Prove that the n × n matrix
h
i
PS←T = [w1 ]S [w2 ]S · · · [wn ]S
satisfies the property
[v]s = PS←T [v]T
4. Exercises §3.7 #1, 2, 4, 14, 15, 36.
5
for all v ∈ V.