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Chapter 3
Section 3.2
Vector Space Properties of ℝ𝑛
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 “vectors”
that satisfy the 2 closure, 4 addition
and 4 multiplication properties given
to the right.
The set of column matrices we called
ℝ𝑛 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 𝑎, 𝑏 ∈ ℝ. W has the following properties.
closure
(c1) x + y ∈ 𝑊
(c2) 𝑎x ∈ 𝑊
Addition
(a1) x + y = y + x
(a2) x + y + z = x + y + z
(a3) 𝜽 ∈ 𝑊 and x + 𝜽 = x for all x
(a4) If x ∈ 𝑊 then −x ∈ 𝑊 and x + −x = 𝜽
Multiplication
(m1) 𝑎 𝑏x = 𝑎𝑏 x
(m2) 𝑎 x + y = 𝑎x + 𝑎y
(m3) 𝑎 + 𝑏 x = 𝑎x + 𝑏x
(m4) 1x = x for all x ∈ 𝑊
ℝ𝑛 =
𝑥1
⋮ : 𝑥1 , ⋯ , 𝑥𝑛 ∈ ℝ is a vector space.
𝑥𝑛
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 + −a = b + −a
3x + a + −a = b − a
3x + 𝜽 = b − a
3x = b − a
1
1
3x
=
b−a
3
3
1
3 x = 13 b − a
3
1x = 13 b − a
x = 13 b − 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 ℝ𝑛
that is itself a vector space is called a
subspace of ℝ𝑛 . Any subset will satisfy
the inherited properties (a1), (a2),
(m1), (m2), (m3), (m4).
Subspace Theorem
A subset W of ℝ𝑛 is a subspace of ℝ𝑛 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 ∈ 𝑊 and 𝑎 ∈ ℝ then 𝑎x ∈ 𝑊
(a3) 𝜽 ∈ 𝑊 and x + 𝜽 = x for all x
(c1) If x,y ∈ 𝑊 then x + y ∈ 𝑊
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 𝑊 =
𝑥1
𝑥2 : 𝑥1 ∙ 𝑥2 > 0 is not a subspace.
The vector 𝜽 =
Example
Show 𝑊 =
0
∉ 𝑊 since 0 ∙ 0 = 0 ≯ 0
0
𝑥1
𝑥2 : 2𝑥1 + 𝑥2 = 0 is a
subspace.
0
1. Show (a3): 2 ∙ 0 + 0 = 0 which means
=𝜽∈𝑊
0
𝑦1
𝑥1
2. Show (c1): If x,y ∈ 𝑊 with x = 𝑥 and y = 𝑦 then 2𝑥1 + 𝑥2 = 0 and 2𝑦1 + 𝑦2 = 0
2
2
𝑥 +𝑦
now x + y = 𝑥1 + 𝑦1
2
2
then 2 𝑥1 + 𝑦1 + 𝑥2 + 𝑦2 = 2𝑥1 + 𝑥2 + 2𝑦1 + 𝑦2 = 0 + 0 = 0
which means x + y ∈ 𝑊
𝑥1
3. Show (c2): If x ∈ 𝑊 and 𝑎 ∈ ℝ with x = 𝑥 then 2𝑥1 + 𝑥2 = 0
2
𝑎𝑥1
now 𝑎x = 𝑎𝑥
2
then 2𝑎𝑥1 + 𝑎𝑥2 = 𝑎 2𝑥1 + 𝑥2 = 𝑎 ∙ 0 = 0 which means 𝑎x ∈ 𝑊
Example
𝑥1
𝑥2 : 𝑥1 ∙ 𝑥2 = 0 is or is not a subspace.
2
0
2
This is not subspace. If x =
and y =
then x,y ∈ 𝑊, but x + y =
∉𝑊
0
3
3
Show that 𝑊 =
Example
Show that 𝑊 = x: x = 𝑥1
2
1
3 + 𝑥2 0 , 𝑥1 , 𝑥2 ∈ ℝ is or is not a subspace.
5
4
This is subspace.
2
0
1
1. Show (a3): 0 3 + 0 0 = 0 = 𝜃 which means 𝜽 ∈ 𝑊
5
0
4
2
2
1
1
2. Show (c1): If x, y ∈ 𝑊 with x = 𝑥1 3 + 𝑥2 0 and y = 𝑦1 3 + 𝑦2 0
5
5
4
4
2
2
2
1
1
then x + y = 𝑥1 3 + 𝑥2 0 + 𝑦1 3 + 𝑦2 0 = 𝑥1 + 𝑦1 3 + 𝑥2 + 𝑦2
5
5
5
4
4
2
1
3. Show (c2): If x ∈ 𝑊 and 𝑎 ∈ ℝ with x = 𝑥1 3 + 𝑥2 0
5
4
2
2
1
1
then 𝑎x = 𝑎 𝑥1 3 + 𝑥2 0 = 𝑎𝑥1 3 + 𝑎𝑥2 0 ∈ 𝑊
5
5
4
4
1
0 ∈𝑊
4
Example
Let A be a 𝑚 × 𝑛 matrix show that 𝑊 = x: 𝐴x = 𝜽 is a subspace of ℝ𝑛 . (We will call this
the kernel of matrix A.)
1. Show (a3): 𝐴𝜽 = 𝜽 which means 𝜽 ∈ 𝑊
2. Show (c1): If x,y ∈ 𝑊 then 𝐴x = 𝜽 and 𝐴y = 𝜽
then 𝐴 x + y = 𝐴x + 𝐴y = 𝜽 + 𝜽 = 𝜽 which means x + y ∈ 𝑊
3. Show (c2): If x ∈ 𝑊 and 𝑎 ∈ ℝ then 𝐴x = 𝜽, then 𝐴 𝑎x = 𝑎 𝐴x = 𝑎𝜽 = 𝜽
which means 𝑎x ∈ 𝑊
Example
Let v be a vector in ℝ𝑛 , show that 𝑊 = x: x 𝑇 v=0 is a subspace of ℝ𝑛 . (We call this the
orthogonal space to the vector v.)
1. Show (a3): 𝜽𝑇 v = 0 which means 𝜽 ∈ 𝑊
2. Show (c1): If x,y ∈ 𝑊 then x 𝑇 v = 0 and y 𝑇 v = 0,
then x + y 𝑇 v = x 𝑇 + y 𝑇 v = x 𝑇 v + y 𝑇 v = 0 + 0 = 0
3. Show (c2): If x ∈ 𝑊and 𝑎 ∈ ℝ then x 𝑇 v = 0 then 𝑎x 𝑇 v = 𝑎 x 𝑇 v = 𝑎 ∙ 0 = 0
which means 𝑎x ∈ 𝑊