Section 3.6 Vector Spaces: The Big Idea
Vector spaces are the primary objects that we study in linear algebra. Generally speaking, a vector space is a collection of objects (called vectors) that preserves linear relationships; that is, the objects work well under vector addition and scalar multiplication. As you will see shortly, vectors are not always going to be the column vectors we have seen so far or viewed geometrically as arrows from one point to another.
Definition 3.6.1.
A vector space, \(V\text{,}\) is a nonempty set of objects called vectors with two operations called addition and scalar multiplication such that the following hold for all \(\vec{u}, \vec{v}, \vec{w} \in V\) and \(c,d \in \mathbb{R}\text{:}\)
If \(\vec{u}, \vec{v} \in V\text{,}\) then \(\vec{u}+\vec{v}\in V\text{.}\)
\(\displaystyle \vec{u}+\vec{v}=\vec{v}+\vec{u}\)
\(\displaystyle (\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})\)
There exists a vector \(\vec{0}_V\) such that \(\vec{v}+\vec{0}_V=\vec{v}\text{.}\)
For each \(\vec{u} \in V\text{,}\) there is a vector \(-\vec{u} \in V\) such that \(\vec{u} + (-\vec{u})=\vec{0}_V\text{.}\)
If \(\vec{u} \in V\) and \(c \in \mathbb{R}\text{,}\) then \(c\vec{u} \in V\text{.}\)
\(\displaystyle c(\vec{u}+\vec{v})=c\vec{u}+c\vec{v}\)
\(\displaystyle (c+d)\vec{v}=c\vec{v}+d\vec{v}\)
\(\displaystyle c(d\vec{v})=(cd)\vec{v}\)
\(\displaystyle 1 \vec{v}=\vec{v}\)
You can refer to these properties as
closure of vector addition
commutativity of vector addition
associativity of vector addition
existence of the zero vector
existence of the additive inverse
closure of scalar multiplication
distributive property of scalar multiplication across vector addition
distributive property of scalar addition across scalar multiplication (of a vector)
associativity of scalar multiplication
existence of scalar multiplicative identity
This is the definition for a real vector space since the scalars come from \(\mathbb{R}\text{,}\) the real numbers. Sometimes it will be useful for us to consider complex vector spaces (scalars come from \(\mathbb{C}\)), but unless otherwise stated, you should assume that you are working with a real vector space.
Investigation 3.6.1.
In order to gain an appreciation of definitions, use only the above definition to prove the following results:
(a)
The zero vector is unique. You can begin this by supposing that there exists some \(\vec{w}\) such that \(\vec{x} +\vec{w} = \vec{x}\) for every \(\vec{x} \in V\text{,}\) then you need to show that \(\vec{w}\) must equal \(\vec{0}_V\text{.}\)
(b)
The additive inverse of a vector is unique.
Example 3.6.2.
The real numbers, \(\mathbb{R}\text{,}\) are a vector space since all of the above properties hold.
Example 3.6.3.
Real valued vectors, \(\mathbb{R}^n\text{,}\) are a vector space since all of the above properties hold when vector addition and scalar multiplication are done componentwise. We can think of the vectors in \(\mathbb{R}^n\) as a directed line segment (an arrow) or as a point in \(n\)-dimensional space.
Investigation 3.6.2.
Show why \(\mathbb{Z}^n\text{,}\) the set of vectors with \(n\) integer components is not a vector space.
Investigation 3.6.3.
Is \(\mathbb{C}^n\) a real vector space? Why or why not?
Investigation 3.6.4.
Is \(\mathbb{R}^n\) a complex vector space? Why or why not?
Example 3.6.4.
The set of
\(m\) by
\(n\) matrices over the real numbers,
\(M_{m \times n}(\mathbb{R})\) or simply
\(M_{m \times n}\text{,}\) is a vector space since all of the above properties hold when “vector” addition and scalar multiplication are done entry wise. The quotes are around vector in the previous sentence because you may not always think of matrices as being vectors but using the properties from
Section 2.2, you can treat matrices as vectors in the general sense.
Investigation 3.6.5.
The set of polynomials (in variable \(t\)) of degree at most \(n\) is denoted by \(\mathbb{P}_n\text{.}\)
(a)
Is \(t^2-4t \in \mathbb{P}_2\text{?}\)
(b)
Is \(3t^2+t \in \mathbb{P}_3\text{?}\)
(c)
Is \(t^2-t+1 \in \mathbb{P}_1\text{?}\)
(d)
Write \(\mathbb{P}_n\) as a set using set builder notation. Be sure you have a statement that you can use to verify if an object is in your set or not.
(e)
Prove that \(\mathbb{P}_n\) is a real vector space.
Example 3.6.5.
The following sets are also vector spaces:
The set of all polynomials (in variable \(t\)) denoted \(\mathbb{P}\text{.}\)
\(F(S,\mathbb{R})\text{,}\) the set of functions from a set \(S\) to the real numbers. We will take a closer look at this vector space in the next problem.
\(\{\vec{0}\}\text{,}\) the trivial vector space.
Investigation 3.6.6.
We are going to take a look at the vector space \(V=F(\{a,b,c\},\mathbb{R})\) to get used to our more general way of thinking about vectors and vector spaces. You should think of the vector space \(V\) as the set of functions with domain \(\{a,b,c\}\) and codomain \(\mathbb{R}\text{.}\) In other words, we are looking at the set of functions that only use \(a\text{,}\) \(b\text{,}\) and \(c\) as inputs and have outputs of real numbers.
Let \(g_1\) be the function that takes \(a\text{,}\) \(b\text{,}\) and \(c\) to \(3\text{,}\) \(-2\text{,}\) and \(0\) respectively.
Let \(g_2\) be the function that takes \(a\text{,}\) \(b\text{,}\) and \(c\) to \(-2\text{,}\) \(7\text{,}\) and \(1\) respectively.
Let \(g_3\) be the function that takes \(a\text{,}\) \(b\text{,}\) and \(c\) to \(1\text{,}\) \(1\text{,}\) and \(1\) respectively.
Let \(g_4\) be the function that takes \(a\text{,}\) \(b\text{,}\) and \(c\) to \(0\text{,}\) \(0\text{,}\) and \(0\) respectively.
(a)
Fill in the blank: \(g_2(b) =\)
Fill in the blank: \(g_3(a) =\)
Fill in the blank: \(g_1(c) =\)
(b)
Does it make sense to add the inputs of these functions? Explain.
(c)
Does it make sense to add the outputs of these functions? Explain.
(d)
Let \(g_5\) be the function that takes \(5\text{,}\) \(1\text{,}\) and \(0\) to \(a\text{,}\) \(b\text{,}\) and \(c\) respectively. Is \(g_5 \in V\text{?}\)
(e)
Describe the function \(g_1 +g_2\text{.}\) In other words, give the outputs for all possible inputs and write a sentence about how you built \(g_1+g_2\) in terms of \(g_1\) and \(g_2\text{.}\)
(f)
Describe the function \(3 g_3\text{.}\)
(g)
What function when added to \(g_2\) will give \(g_4\text{?}\)
(h)
Can you write \(g_1\) as a linear combination of the set \(\{ g_2 , g_3 , g_4\}\text{?}\) Explain why or why not.
(i)
Can you write \(g_4\) as a linear combination of the set \(\{ g_2 , g_3 , g_1\}\text{?}\) Explain why or why not.
Investigation 3.6.7.
(a)
Write a sentence or two about what property makes a vector \(\vec{v}
\in V\) the zero vector for \(V\text{,}\) called \(\vec{0}_V\text{.}\)
(b)
What is the zero vector for the vector space \(M_{m \times
n}\text{?}\) Remember to state your answer as an element of \(M_{m
\times n}\text{.}\)
(c)
What is the zero vector for the vector space \(\mathbb{P}_n\text{?}\) Remember to state your answer as an element of \(\mathbb{P}_n\text{.}\)
(d)
What is the zero vector for the vector space \(\mathbb{P}\text{?}\) Remember to state your answer as an element of \(\mathbb{P}\text{.}\)
(e)
What is the zero vector for the vector space \(F(\mathbb{R},\mathbb{R})\text{?}\) Remember to state your answer as an element of \(F(\mathbb{R},\mathbb{R})\text{.}\)
