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Chapter 5 Chapter Content
1. Real Vector Spaces
2. Subspaces
3. Linear Independence
4. Basis and Dimension
5. Row Space, Column Space, and Nullspace
6. Rank and Nullity
Definition (Vector Space)
Let V be an arbitrary nonempty set of objects on which two operations are
defined: addition, and multiplication by scalars. If the following axioms are
satisfied by all objects u, v, w in V and all scalars k and m, then we call V a
vector space and we call the objects in V vectors
1. If u and v are objects in V, then u + v is in V.
2. u + v = v + u
3. u + (v + w) = (u + v) + w
4. There is an object 0 in V, called a zero vector for V, such that 0 + u= u + 0 = u
for all u in V.
5. For each u in V, there is an object -u in V, called a negative of u, such that
u + (-u) = (-u) + u = 0.
6. If k is any scalar and u is any object in V, then ku is in V.
7. k (u + v) = ku + kv
8. (k + m) u = ku + mu
9. k (mu) = (km) (u)
10. 1u = u
Remarks
• Depending on the application, scalars may be real numbers or complex
numbers. Vector spaces in which the scalars are complex numbers are called
complex vector spaces, and those in which the scalars must be real are called
real vector spaces.
• The definition of a vector space specifies neither the nature of
the vectors nor the operations. Any kind of object can be a vector, and the
operations of addition and scalar multiplication may not have any relationship or
similarity to the standard vector operations on R n .
• The only requirement is that the ten vector space axioms be satisfied.
Example ( R n Is a Vector Space)
The set V = R n with the standard operations of addition and scalar multiplication
is a vector space. (Axioms 1 and 6 follow from the definitions of the
standard operations on R n ; the remaining axioms follow from Theorem 4.1.1.)
The three most important special cases of R n are R (the real numbers), R 2
(the vectors in the plane), and R 3 (the vectors in 3-space).
Example (2×2 Matrices)
Show that the set V of all 2×2 matrices with real entries is a vector space if
vector addition is defined to be matrix addition and vector scalar
multiplication is defined to be matrix scalar multiplication.
 u11 u12 
 v11 v12 
and
v


v

u21 u22 
 21 v22 
Solution: Let u  
(1) we must show that u + v is a 2×2 matrix.
 u11  v11 u12  v12 
uv  

u

v
u

v
 21 21 22 22 
(2) Want to show that u + v = v + u
 u11  v11 u12  v12 
uv  
  vu
u

v
u

v
 21 21 22 22 
(3) Similarly we can show that u + ( v + w ) = ( u + v )+ w.
(4) Define 0 to be
0 0 such that 0  u   u11 u12   u  0  u
0
u


u
0
0

21
22



 u11 u12 

u

(5) Define the negative of u to be
u
 such that

u
 21
22 
0 0
u  u  u  u  

0
0


 ku11 ku12 
ku

(6) If k is any scalar and u is a 2X2 matrix, then
ku
 is 2X2 matrix.
ku
 21
22 
(7)-(9) will be obtained by similar approach.
1u11 1u12   u11 u12 
  u
  u.
1
u
1
u
u
 21
22 
 21 22 
(10) 1u  
Thus, the set V of all 2×2 matrices with real entries is a vector space.
Example: Given the set of all triples of real numbers ( x, y, z ) with the operations
( x, y, z )  ( x ', y ', z ')  ( x  x ', y  y ', z  z ') and
k ( x, y, z )  (kx, y, z )
Show that it’s a vector space under the given operation.
Solution: We must check all ten properties:
(1) If (x, y, z) and (x’, y’, z’) are triples of real numbers, so is
(x, y, z) + (x’, y’, z’) = (x + x’, y +y’, z + z’).
(2) (x, y, z) + (x’, y’, z’) = (x + x’, y + y’, z + z’)= (x’, y’, z’) + (x, y, z).
(3) (x, y, z) + [(x’, y’, z’) + (x’’, y’’, z’’)] = (x, y, z) + [(x’, y’, z’) + (x’’, y’’, z’’)].
(4) There is an object 0, (0, 0, 0), such that
(0, 0, 0) + (x, y, z) = (x, y, z) + (0, 0, 0)= (x, y, z).
(5) For each positive real x, (-x, -y, -z) acts as the negative:
(x, y, z) + (-x, -y, -z) = (-x, -y, -z) + (x, y, z) =(x, y, z)
(6) If k is a real and (x, y, z) is a triple of real numbers, then k (x, y, z) = (kx,
y, z) is again a triple of real numbers.
(7) k[(x, y, z) + (x’, y’, z’)] = (k(x+x’), y+y’, z+z’) = k(x, y, z) + k(x’, y’, z’)
(8) (k + m) (x, y, z) = ((k + m)x, y, z) = k (x, y, z) + m(x, y, z)
(9) k(m(x, y, z)) = (kmx, y, z) = (km)(x, y, z)
(10) 1 (x, y, z) = (x, y, z)
Thus, the set of all triples of real numbers ( x, y, z ) with the operations is a
vector space under the given operation.
Example (Not a Vector Space)
Let V = R2 and define addition and scalar multiplication operations as follows:
If u = (u1, u2) and v = (v1, v2), then define
u + v = (u1 + v1, u2 + v2)
and if k is any real number, then define
k u = (k u1, 0)
There are values of u for which Axiom 10 fails to hold. For example, if u = (u1,
u2) is such that u2 ≠ 0,then
1u = 1 (u1, u2) = (1 u1, 0) = (u1, 0) ≠ u
Thus, V is not a vector space with the stated operations
The Zero Vector Space
Let V consist of a single object, which we denote by 0, and define
0 + 0 = 0 and k 0 = 0 for all scalars k.
It’s easy to check that all the vector space axioms are satisfied.
We called this the zero vector space.
Theorem 5.1.1
Let V be a vector space, u be a vector in V, and k a scalar; then:
(a) 0 u = 0
(b) k 0 = 0
(c) (-1) u = -u
(d) If k u = 0 , then k = 0 or u = 0.
5.2 Subspaces
Definition
A subset W of a vector space V is called a subspace of V if W is itself a
vector space under the addition and scalar multiplication defined on V.
Theorem 5.2.1
If W is a set of one or more vectors from a vector space V, then W is a
subspace of V if and only if the following conditions hold:
a) If u and v are vectors in W, then u + v is in W.
b) If k is any scalar and u is any vector in W , then ku is in W.
Remark
Theorem 5.2.1 states that W is a subspace of V if and only if W is a closed under
addition (condition (a)) and closed under scalar multiplication (condition (b)).
Example
All vectors of the form (a, 0, 0) is a subspace of R3.
• The set is closed under vector addition because
(a, 0, 0) + (b, 0, 0) = (a + b, 0, 0)
• It is closed under scalar multiplication because
k(a, 0, 0) = (ka, 0, 0)
Therefore it is a subspace of R3.
Example (Not a Subspace)
Let W be the set of all points (x, y) in R2 such that x ≥ 0 and y ≥ 0. These are the
points in the first quadrant.
The set W is not a subspace of R2 since it is not closed under scalar multiplication.
For example, v = (1, 1) lines in W, but its negative (-1)v = -v = (-1, -1) does not.
Subspaces of Mnn
The set of n×n diagonal matrices forms subspaces of Mnn, since each of these
sets is closed under addition and scalar multiplication.
The set of n×n matrices with integer entries is NOT a subspace of the vector space
Mnn of n×n matrices.
This set is closed under vector addition since the sum of two integers is again an
integer. However, it is not closed under scalar multiplication since the product ku
where k is real and a is an integer need not be an integer. Thus, the set is not a
subspace.
Solution Space
Solution Space of Homogeneous Systems
If Ax = b is a system of the linear equations, then each vector x that satisfies this
equation is called a solution vector of the system.
Theorem 5.2.2
If Ax = 0 is a homogeneous linear system of m equations in n unknowns, then
the set of solution vectors is a subspace of Rn.
Remark: Theorem 5.2.2 shows that the solution vectors of a homogeneous
linear system form a vector space, which we shall call the solution space of the
system.
Linear Combination
Definition
A vector w is a linear combination of the vectors v1, v2,…, vr if it can be
expressed in the form
w = k1v1 + k2v2 + · · · + kr vr
where k1, k2, …, kr are scalars.
Example:Vectors in R3 are linear combinations of i, j, and k
Every vector v = (a, b, c) in R3 is expressible as a linear combination of the
standard basis vectors
i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)
Since
v= a(1, 0, 0) + b(0, 1, 0) + c(0, 0, 1) = a i + b j + c k
Example
Consider the vectors u = (1, 2, -1) and v = (6, 4, 2) in R3. Show that w = (9, 2,
7) is a linear combination of u and v and that w′ = (4, -1, 8) is not a linear
combination of u and v.
Solution.
In order for w to be a linear combination of u and v, there must be scalars k1
and k2 such that w = k1u + k2v;
(9, 2, 7) = (k1 + 6k2, 2k1 + 4k2, -k1 + 2k2)
Equating corresponding components gives
k1 + 6k2 = 9
2k1+ 4k2 = 2
-k1 + 2k2 = 7
Solving this system yields k1 = -3, k2 = 2, so
w = -3u + 2v
Similarly, for w‘ to be a linear combination of u and v, there must be scalars k1
and k2 such that w'= k1u + k2v;
(4, -1, 8) = k1(1, 2, -1) + k2(6, 4, 2)
or
(4, -1, 8) = (k1 + 6k2, 2k1 + 4k2, -k1 + 2k2)
Equating corresponding components gives
k1 + 6k2 = 4
2 k1+ 4k2 = -1
- k1 + 2k2 = 8
This system of equation is inconsistent, so no such scalars k1 and k2 exist.
Consequently, w' is not a linear combination of u and v.
Linear Combination and Spanning
Theorem 5.2.3
If v1, v2, …, vr are vectors in a vector space V, then:
(a) The set W of all linear combinations of v1, v2, …, vr is a subspace of V.
(b) W is the smallest subspace of V that contain v1, v2, …, vr in the sense that
every other subspace of V that contain v1, v2, …, vr must contain W.
Definition
If S = {v1, v2, …, vr} is a set of vectors in a vector space V, then the subspace
W of V containing of all linear combination of these vectors in S is called the
space spanned by v1, v2, …, vr, and we say that the vectors v1, v2, …, vr
span W. To indicate that W is the space spanned by the vectors in the set S
= {v1, v2, …, vr}, we write
W = span (S) or W = span{v1, v2, …, vr}.
Example
If v1 and v2 are non-collinear vectors in R3 with their initial points at
the origin, then span{v1, v2}, which consists of all linear combinations
k1v1 + k2v2 is the plane determined by v1 and v2.
Similarly, if v is a nonzero vector in R2 and R3, then span {v}, which
is the set of all scalar multiples kv, is the line determined by v.
Example
Determine whether v1 = (1, 1, 2), v2 = (1, 0, 1), and v3 = (2, 1, 3)
span the vector space R3.
Solution
Is it possible that an arbitrary vector b = (b1, b2, b3) in R3 can be expressed as
a linear combination b = k1v1 + k2v2 + k3v3 ?
b = (b1, b2, b3) = k1(1, 1, 3) + k2(1, 0, 1) + k3(2, 1, 3)
= (k1+k2+2k3, k1+k3, 2k1+k2+3k3)
Or
k1 + k2 + 2k3 = b1
k1 + k3 = b2
2k1 + k2 + 3 k3 = b3
This system is consistent for all values of b1, b2, and b3 if and only if the
coefficient matrix has a nonzero determinant.
However, det(A) = 0, so that v1, v2, and v3, do not span R3.
Theorem 5.2.4
If S = {v1, v2, …, vr} and S′ = {w1, w2, …, wr} are two sets of vector in a vector
space V, then
span{v1, v2, …, vr} = span{w1, w2, …, wr}
if and only if each vector in S is a linear combination of these in S′ and each
vector in S′ is a linear combination of these in S.