Transcript Chapter 3

Chapter 3
Section 3.2
Vector Space Properties of โ„๐‘›
What is a Vector?
The answer to this questions depends
on who you ask. There are several
ways to vectors.
(Physicist-geometric) Anything that
has been assigned a direction and
length.
(Computer Scientist-numeric) An
ordered list of numbers.
(Mathematician-algebraic) An element
in a vector space.
A vector space is a set of โ€œvectorsโ€
that satisfy the 2 closure, 4 addition
and 4 multiplication properties given
to the right.
The set of column matrices we called
โ„๐‘› is a vector space due to the
properties of adding and scalar
multiplication of matrices.
Vector Space Properties
Let x,y, and z be vectors in a vector space W
and ๐‘Ž, ๐‘ โˆˆ โ„. W has the following properties.
closure
(c1) x + y โˆˆ ๐‘Š
(c2) ๐‘Žx โˆˆ ๐‘Š
Addition
(a1) x + y = y + x
(a2) x + y + z = x + y + z
(a3) ๐œฝ โˆˆ ๐‘Š and x + ๐œฝ = x for all x
(a4) If x โˆˆ ๐‘Š then โˆ’x โˆˆ ๐‘Š and x + โˆ’x = ๐œฝ
Multiplication
(m1) ๐‘Ž ๐‘x = ๐‘Ž๐‘ x
(m2) ๐‘Ž x + y = ๐‘Žx + ๐‘Žy
(m3) ๐‘Ž + ๐‘ x = ๐‘Žx + ๐‘x
(m4) 1x = x for all x โˆˆ ๐‘Š
โ„๐‘› =
๐‘ฅ1
โ‹ฎ : ๐‘ฅ1 , โ‹ฏ , ๐‘ฅ๐‘› โˆˆ โ„ is a vector space.
๐‘ฅ๐‘›
Algebra and Vectors
The properties of a vector
space are the fundamental
concepts needed in order to do
basic algebraic manipulations
like solving equations. The
example to the right show how
the properties apply to solving
3x + a = b
3x + a = b
3x + a + โˆ’a = b + โˆ’a
3x + a + โˆ’a = b โˆ’ a
3x + ๐œฝ = b โˆ’ a
3x = b โˆ’ a
1
1
3x
=
bโˆ’a
3
3
1
3 x = 13 b โˆ’ a
3
1x = 13 b โˆ’ a
x = 13 b โˆ’ a
(c1)
(a1)
(a4)
(a3)
(c2)
(m1)
(m4)
Subspaces
A subset W of vectors may or may not
form a vector space. A subset W of โ„๐‘›
that is itself a vector space is called a
subspace of โ„๐‘› . Any subset will satisfy
the inherited properties (a1), (a2),
(m1), (m2), (m3), (m4).
Subspace Theorem
A subset W of โ„๐‘› is a subspace of โ„๐‘› if and
only if W satisfies the following 3 properties:
The theorem to the right shows exactly
when a subset is a subspace.
(c2) If x โˆˆ ๐‘Š and ๐‘Ž โˆˆ โ„ then ๐‘Žx โˆˆ ๐‘Š
(a3) ๐œฝ โˆˆ ๐‘Š and x + ๐œฝ = x for all x
(c1) If x,y โˆˆ ๐‘Š then x + y โˆˆ ๐‘Š
Showing W is not a subspace
To show W is not a subspace you
need to give a specific example of
how W does not satisfy one of the
properties.
Showing W is a subspace.
To show a subset W is a subspace you
need to show that W satisfies all 3
conditions of the subspaces theorem.
Example
Show ๐‘Š =
๐‘ฅ1
๐‘ฅ2 : ๐‘ฅ1 โˆ™ ๐‘ฅ2 > 0 is not a subspace.
The vector ๐œฝ =
Example
Show ๐‘Š =
0
โˆ‰ ๐‘Š since 0 โˆ™ 0 = 0 โ‰ฏ 0
0
๐‘ฅ1
๐‘ฅ2 : 2๐‘ฅ1 + ๐‘ฅ2 = 0 is a
subspace.
0
1. Show (a3): 2 โˆ™ 0 + 0 = 0 which means
=๐œฝโˆˆ๐‘Š
0
๐‘ฆ1
๐‘ฅ1
2. Show (c1): If x,y โˆˆ ๐‘Š with x = ๐‘ฅ and y = ๐‘ฆ then 2๐‘ฅ1 + ๐‘ฅ2 = 0 and 2๐‘ฆ1 + ๐‘ฆ2 = 0
2
2
๐‘ฅ +๐‘ฆ
now x + y = ๐‘ฅ1 + ๐‘ฆ1
2
2
then 2 ๐‘ฅ1 + ๐‘ฆ1 + ๐‘ฅ2 + ๐‘ฆ2 = 2๐‘ฅ1 + ๐‘ฅ2 + 2๐‘ฆ1 + ๐‘ฆ2 = 0 + 0 = 0
which means x + y โˆˆ ๐‘Š
๐‘ฅ1
3. Show (c2): If x โˆˆ ๐‘Š and ๐‘Ž โˆˆ โ„ with x = ๐‘ฅ then 2๐‘ฅ1 + ๐‘ฅ2 = 0
2
๐‘Ž๐‘ฅ1
now ๐‘Žx = ๐‘Ž๐‘ฅ
2
then 2๐‘Ž๐‘ฅ1 + ๐‘Ž๐‘ฅ2 = ๐‘Ž 2๐‘ฅ1 + ๐‘ฅ2 = ๐‘Ž โˆ™ 0 = 0 which means ๐‘Žx โˆˆ ๐‘Š
Example
๐‘ฅ1
๐‘ฅ2 : ๐‘ฅ1 โˆ™ ๐‘ฅ2 = 0 is or is not a subspace.
2
0
2
This is not subspace. If x =
and y =
then x,y โˆˆ ๐‘Š, but x + y =
โˆ‰๐‘Š
0
3
3
Show that ๐‘Š =
Example
Show that ๐‘Š = x: x = ๐‘ฅ1
2
1
3 + ๐‘ฅ2 0 , ๐‘ฅ1 , ๐‘ฅ2 โˆˆ โ„ is or is not a subspace.
5
4
This is subspace.
2
0
1
1. Show (a3): 0 3 + 0 0 = 0 = ๐œƒ which means ๐œฝ โˆˆ ๐‘Š
5
0
4
2
2
1
1
2. Show (c1): If x, y โˆˆ ๐‘Š with x = ๐‘ฅ1 3 + ๐‘ฅ2 0 and y = ๐‘ฆ1 3 + ๐‘ฆ2 0
5
5
4
4
2
2
2
1
1
then x + y = ๐‘ฅ1 3 + ๐‘ฅ2 0 + ๐‘ฆ1 3 + ๐‘ฆ2 0 = ๐‘ฅ1 + ๐‘ฆ1 3 + ๐‘ฅ2 + ๐‘ฆ2
5
5
5
4
4
2
1
3. Show (c2): If x โˆˆ ๐‘Š and ๐‘Ž โˆˆ โ„ with x = ๐‘ฅ1 3 + ๐‘ฅ2 0
5
4
2
2
1
1
then ๐‘Žx = ๐‘Ž ๐‘ฅ1 3 + ๐‘ฅ2 0 = ๐‘Ž๐‘ฅ1 3 + ๐‘Ž๐‘ฅ2 0 โˆˆ ๐‘Š
5
5
4
4
1
0 โˆˆ๐‘Š
4
Example
Let A be a ๐‘š × ๐‘› matrix show that ๐‘Š = x: ๐ดx = ๐œฝ is a subspace of โ„๐‘› . (We will call this
the kernel of matrix A.)
1. Show (a3): ๐ด๐œฝ = ๐œฝ which means ๐œฝ โˆˆ ๐‘Š
2. Show (c1): If x,y โˆˆ ๐‘Š then ๐ดx = ๐œฝ and ๐ดy = ๐œฝ
then ๐ด x + y = ๐ดx + ๐ดy = ๐œฝ + ๐œฝ = ๐œฝ which means x + y โˆˆ ๐‘Š
3. Show (c2): If x โˆˆ ๐‘Š and ๐‘Ž โˆˆ โ„ then ๐ดx = ๐œฝ, then ๐ด ๐‘Žx = ๐‘Ž ๐ดx = ๐‘Ž๐œฝ = ๐œฝ
which means ๐‘Žx โˆˆ ๐‘Š
Example
Let v be a vector in โ„๐‘› , show that ๐‘Š = x: x ๐‘‡ v=0 is a subspace of โ„๐‘› . (We call this the
orthogonal space to the vector v.)
1. Show (a3): ๐œฝ๐‘‡ v = 0 which means ๐œฝ โˆˆ ๐‘Š
2. Show (c1): If x,y โˆˆ ๐‘Š then x ๐‘‡ v = 0 and y ๐‘‡ v = 0,
then x + y ๐‘‡ v = x ๐‘‡ + y ๐‘‡ v = x ๐‘‡ v + y ๐‘‡ v = 0 + 0 = 0
3. Show (c2): If x โˆˆ ๐‘Šand ๐‘Ž โˆˆ โ„ then x ๐‘‡ v = 0 then ๐‘Žx ๐‘‡ v = ๐‘Ž x ๐‘‡ v = ๐‘Ž โˆ™ 0 = 0
which means ๐‘Žx โˆˆ ๐‘Š