Learning Outcomes
- Explain why a given set with defined addition and scalar multiplication does satisfy a given vector space property, but nonetheless isn't a vector space.
Section 2.1 Vector Spaces (VS1)
Subsection 2.1.1 Class Activities
Observation 2.1.1.
Several properties of the real numbers, such as commutivity:
\begin{equation*}
x + y = y + x
\end{equation*}
also hold for Euclidean vectors with multiple components:
\begin{equation*}
\left[\begin{array}{c}x_1\\x_2\end{array}\right]
+
\left[\begin{array}{c}y_1\\y_2\end{array}\right]
=
\left[\begin{array}{c}y_1\\y_2\end{array}\right]
+
\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}
\end{equation*}
Activity 2.1.2.
Consider each of the following properties of the real numbers \(\IR^1\text{.}\) Label each property as valid if the property also holds for two-dimensional Euclidean vectors \(\vec u,\vec v,\vec w\in\IR^2\) and numbers \(a,b\in\IR\text{,}\) and invalid if it does not.
- \(\vec u+(\vec v+\vec w)= (\vec u+\vec v)+\vec w\text{.}\)
- \(\vec u+\vec v= \vec v+\vec u\text{.}\)
- There exists some \(\vec z\) where \(\vec v +\vec z =\vec v\text{.}\)
- There exists some \(-\vec v\) where \(\vec v+(-\vec v)=\vec z\text{.}\)
- If \(\vec u\not=\vec v\text{,}\) then \(\frac{1}{2}(\vec u +\vec v )\) is the only vector equally distant from both \(\vec u\) and \(\vec v\)
- \(a(b\vec v)=(ab)\vec v\text{.}\)
- \(1\vec v=\vec v\text{.}\)
- If \(\vec u\not=\vec 0\text{,}\) then there exists some number \(c\) such that \(c\vec u=\vec v\text{.}\)
- \(a(\vec u+\vec v)=a\vec u+a\vec v\text{.}\)
- \((a+b)\vec v=a\vec v+b\vec v\text{.}\)
Definition 2.1.3.
A vector space \(V\) is any set of mathematical objects, called vectors, and a set of numbers, called scalars, with associated addition \(\oplus\) and scalar multiplication \(\odot\) operations that satisfy the following properties. Let \(\vec u,\vec v,\vec w\) be vectors belonging to \(V\text{,}\) and let \(a,b\) be scalars.
- Vector addition is associative: \(\vec u\oplus (\vec v\oplus \vec w)= (\vec u\oplus \vec v)\oplus \vec w\text{.}\)
- Vector addition is commutative: \(\vec u\oplus \vec v= \vec v\oplus \vec u\text{.}\)
- An additive identity exists: There exists some \(\vec z\) where \(\vec v\oplus \vec z=\vec v\text{.}\)
- Additive inverses exist: There exists some \(-\vec v\) where \(\vec v\oplus (-\vec v)=\vec z\text{.}\)
- Scalar multiplication is associative: \(a\odot(b\odot\vec v)=(ab)\odot\vec v\text{.}\)
- 1 is a multiplicative identity: \(1\odot\vec v=\vec v\text{.}\)
- Scalar multiplication distributes over vector addition: \(a\odot(\vec u\oplus \vec v)=(a\odot\vec u)\oplus(a\odot\vec v)\text{.}\)
- Scalar multiplication distributes over scalar addition: \((a+ b)\odot\vec v=(a\odot\vec v)\oplus(b\odot \vec v)\text{.}\)
Remark 2.1.4.
In this text, we only consider real vector spaces, i.e. those whose scalars come from \(\IR\text{.}\) However, one can similarly define vector spaces with scalars from other fields like the complex or rational numbers.
Observation 2.1.5.
Every Euclidean vector space
\begin{equation*}
\IR^n=\setBuilder{\left[\begin{array}{c}x_1\\x_2\\\vdots\\x_n\end{array}\right]}{x_1,x_2,\dots,x_n\in\IR}
\end{equation*}
satisfies all eight requirements for the usual definitions of addition and scalar multiplication, but we will also study other types of vector spaces.
Observation 2.1.6.
The space of \(m \times n\) matrices
\begin{equation*}
M_{m,n}=\setBuilder{\left[\begin{array}{cccc}a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right]}
{a_{11},\ldots,a_{mn} \in\IR}
\end{equation*}
satisfies all eight requirements for component-wise addition and scalar multiplication.
Remark 2.1.7.
Every Euclidean space \(\IR^n\) is a vector space, but there are other examples of vector spaces as well.
For example, consider the set \(\IC\) of complex numbers with the usual defintions of addition and scalar multiplication, and let \(\vec u=a+b\mathbf{i}\text{,}\) \(\vec v=c+d\mathbf{i}\text{,}\) and \(\vec w=e+f\mathbf{i}\text{.}\) Then
\begin{align*}
\vec u+(\vec v+\vec w)
&=
(a+b\mathbf{i})+((c+d\mathbf{i})+(e+f\mathbf{i}))\\
&=
(a+b\mathbf{i})+((c+e)+(d+f)\mathbf{i})
\\&=(a+c+e)+(b+d+f)\mathbf{i}
\\&=((a+c)+(b+d)\mathbf{i})+(e+f\mathbf{i})\\
&=
(\vec u+\vec v)+\vec w
\end{align*}
All eight properties can be verified in this way.
Remark 2.1.8.
The following sets are just a few examples of vector spaces, with the usual/natural operations for addition and scalar multiplication.
- \(\IR^n\text{:}\) Euclidean vectors with \(n\) components.
- \(\IC\text{:}\) Complex numbers.
- \(M_{m,n}\text{:}\) Matrices of real numbers with \(m\) rows and \(n\) columns.
- \(\P_n\text{:}\) Polynomials of degree \(n\) or less.
- \(\P\text{:}\) Polynomials of any degree.
- \(C(\IR)\text{:}\) Real-valued continuous functions.
Activity 2.1.9.
Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\text{.}\)
Which of the following vectors is not in \(V\text{?}\)
- \(\displaystyle (0, 0)\)
- \(\displaystyle (1, 2)\)
- \(\displaystyle (2, 4)\)
- \(\displaystyle (3, 8)\)
Activity 2.1.10.
Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with the operation \(\oplus\) defined by
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2) \text{.}
\end{equation*}
Let \(\vec u, \vec v\) be in \(V\) with \(\vec u=(1, 2)\) and \(\vec v=(2, 4)\text{.}\) Using the operations defined for \(V\text{,}\) which of the following is \(\vec u\oplus\vec v\text{?}\)
- \(\displaystyle (2, 6)\)
- \(\displaystyle (2, 8)\)
- \(\displaystyle (3, 6)\)
- \(\displaystyle (3, 8)\)
Activity 2.1.11.
Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2)
\hspace{3em}
c\odot (x,y)=(cx,y^c)\text{.}
\end{equation*}
Let \(a=2, b=-3\) be scalars and \(\vec u=(1,2) \in V\text{.}\)
(a)
Verify that
\begin{equation*}
(a+b)\odot \vec u=\left(-1,\frac{1}{2}\right)\text{.}
\end{equation*}
(b)
Compute the value of
\begin{equation*}
\left(a\odot \vec u\right)\oplus \left(b\odot \vec u\right)\text{.}
\end{equation*}
Activity 2.1.12.
Consider the set \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2)
\hspace{3em}
c\odot (x,y)=(cx,y^c)\text{.}
\end{equation*}
Let \(a, b\) be unspecified scalars in \(\mathbb R\) and \(\vec u = (x,y)\) be an unspecified vector in \(V\text{.}\)
(a)
Show that both sides of the equation
\begin{equation*}
(a+b)\odot (x,y)=
\left(a\odot (x,y)\right)\oplus \left(b\odot (x,y)\right)
\end{equation*}
simplify to the expression \((ax+bx,y^ay^b)\text{.}\)
(b)
Which of the properties from Definition 2.1.3 did we verify in the previous task?
- Vector addition is associative
- \(1\) is a multiplicative identity
- Scalar multiplication distributes over scalar addition
(c)
Show that \(V\) contains an additive identity element \(\vec{z}=(\unknown,\unknown)\) satisfying
\begin{equation*}
(x,y)\oplus(\unknown,\unknown)=(x,y)
\end{equation*}
for all \((x,y)\in V\) by choosing appropriate values for \(\vec{z}=(\unknown,\unknown)\) and using those to simplify \((x,y)\oplus(\unknown,\unknown)=(x,y)\) to \((x,y)\text{.}\)
Remark 2.1.13.
It turns out \(V=\setBuilder{(x,y)}{y=2^x}\) with operations \(\oplus,\odot\) defined by
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1y_2)
\hspace{3em}
c\odot (x,y)=(cx,y^c)
\end{equation*}
satisifes all eight properties from Definition 2.1.3.
Thus, \(V\) is a vector space.
Activity 2.1.14.
Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+y_1+x_2+y_2,x_1^2+x_2^2)
\end{equation*}
\begin{equation*}
c\odot (x,y)=(x^c,y+c-1)\text{.}
\end{equation*}
(a)
Show that \(1\) is the scalar multiplication identity element by simplifying \(1\odot(x,y)\) to \((x,y)\text{.}\)
(b)
Show that \(V\) does not have an additive identity element \(\vec z=(z,w)\) by showing that \((0,-1)\oplus(z,w)\not=(0,-1)\) for any possible values of \(z,w\text{.}\)
(c)
Is \(V\) a vector space according to Definition 2.1.3?
Activity 2.1.15.
Let \(V=\setBuilder{(x,y)}{x,y\in\IR}\) have operations defined by
\begin{equation*}
(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2,y_1+3y_2)
\hspace{3em}
c\odot (x,y)=(cx,cy)
.
\end{equation*}
(a)
Show that scalar multiplication distributes over vector addition, i.e.
\begin{equation*}
c \odot \left( (x_1,y_1) \oplus (x_2,y_2) \right) = c\odot (x_1,y_1) \oplus c\odot (x_2,y_2)
\end{equation*}
for all \(c\in \IR,\, (x_1,y_1),(x_2,y_2) \in V\text{.}\)
(b)
Show that vector addition is not associative, i.e.
\begin{equation*}
(x_1,y_1) \oplus \left((x_2,y_2) \oplus (x_3,y_3)\right) \neq \left((x_1,y_1)\oplus (x_2,y_2)\right) \oplus (x_3,y_3)
\end{equation*}
for some vectors \((x_1,y_1), (x_2,y_2), (x_3,y_3) \in V\text{.}\)
(c)
Is \(V\) a vector space?
Subsection 2.1.2 Videos
Subsection 2.1.3 Slideshow
Slideshow of activities available at
https://teambasedinquirylearning.github.io/linear-algebra/2023e/VS1.slides.html.Exercises 2.1.4 Exercises
Exercises available at
https://teambasedinquirylearning.github.io/linear-algebra/2023e/exercises/#/bank/VS1/.Subsection 2.1.5 Mathematical Writing Explorations
Exploration 2.1.16.
- Show that \(\mathbb{R}^+\text{,}\) the set of positive real numbers, is a vector space, but where \(x\oplus y\) really means the product (so \(2 \oplus 3 = 6\)), and where scalar multiplication \(\alpha\odot x\) really means \(x^\alpha\text{.}\) Yes, you really do need to check all of the properties, but this is the only time I'll make you do so. Remember, examples aren't proofs, so you should start with arbitrary elements of \(\mathbb R^+\) for your vectors. Make sure you're careful about telling the reader what \(\alpha\) means.
- Prove that the additive identity \(\vec{z}\) in an arbitrary vector space is unique.
- Prove that additive inverses are unique. Assume you have a vector space \(V\) and some \(\vec{v} \in V\text{.}\) Further, assume \(\vec{w_1},\vec{w_2} \in V\) with \(\vec{v} \oplus \vec{w_1} = \vec{v} \oplus \vec{w_2} = \vec{z}\text{.}\) Prove that \(\vec{w_1} = \vec{w_2}\text{.}\)
Exploration 2.1.17.
Consider the vector space of polynomials, \(\P_n\text{.}\) Suppose further that \(n= ab\text{,}\) where \(a \mbox{ and } b\) are each positive integers. Conjecture a relationship between \(M_{a,b}\) and \(\P_n\text{.}\) We will investigate this further in section Section 2.8
Subsection 2.1.6 Sample Problem and Solution
Sample problem Example B.1.5.
