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 20.5 Vector Spaces (AT5)
Subsection 20.5.1 Warm Up
Activity 20.88.
(a)
How would you describe a sandwich to someone who has never seen a sandwich before?
(b)
How would you describe to someone what a vector is?
Subsection 20.5.2 Class Activities
Observation 20.89.
Consider the following applications of properties of the real numbers \(\mathbb R\text{:}\)
- \(1+(2+3)=(1+2)+3\text{.}\)
- \(7+4=4+7\text{.}\)
- There exists some \(\unknown\) where \(5+\unknown=5\text{.}\)
- There exists some \(\unknown\) where \(9+\unknown=0\text{.}\)
- \(\frac{1}{2}(1+7)\) is the only number that is equally distant from \(1\) and \(7\text{.}\)
Activity 20.90.
Which of the following properties of \(\IR^2\) Euclidean vectors is NOT true?
- \(\left[\begin{array}{c} x_1\\x_2\end{array}\right] +\left(\left[\begin{array}{c} y_1\\y_2\end{array}\right] +\left[\begin{array}{c} z_1\\z_2\end{array}\right]\right)= \left(\left[\begin{array}{c} x_1\\x_2\end{array}\right] +\left[\begin{array}{c} y_1\\y_2\end{array}\right]\right) +\left[\begin{array}{c} z_1\\z_2\end{array}\right]\text{.}\)
- \(\left[\begin{array}{c}x_1\\x_2\end{array}\right] + \left[\begin{array}{c}y_1\\y_2\end{array}\right] = \left[\begin{array}{c}y_1\\y_2\end{array}\right] + \left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
- There exists some \(\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]\) where \(\left[\begin{array}{c}x_1\\x_2\end{array}\right] +\left[\begin{array}{c}\unknown\\\unknown\end{array}\right] =\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
- There exists some \(\left[\begin{array}{c}\unknown\\\unknown\end{array}\right]\) where \(\left[\begin{array}{c}x_1\\x_2\end{array}\right]+ \left[\begin{array}{c}\unknown\\\unknown\end{array}\right]= \left[\begin{array}{c}0\\0\end{array}\right]\text{.}\)
- \(\displaystyle\frac{1}{2}\left(\left[\begin{array}{c}x_1\\x_2\end{array}\right] + \left[\begin{array}{c}y_1\\y_2\end{array}\right] \right)\) is the only vector whose endpoint is equally distant from the endpoints of \(\left[\begin{array}{c}x_1\\x_2\end{array}\right]\) and \(\left[\begin{array}{c}y_1\\y_2\end{array}\right]\text{.}\)
Observation 20.91.
Consider the following applications of properties of the real numbers \(\mathbb R\text{:}\)
- \(3(2(7))=(3\cdot 2)(7)\text{.}\)
- \(1(19)=19\text{.}\)
- There exists some \(\unknown\) such that \(\unknown \cdot 4= 9\text{.}\)
- \(3\cdot (2+8)=3\cdot 2+3\cdot 8\text{.}\)
- \((2+7)\cdot 4=2\cdot 4+7\cdot 4\text{.}\)
Activity 20.92.
Which of the following properties of \(\IR^2\) Euclidean vectors is NOT true?
- \(a\left(b\left[\begin{array}{c}x_1\\x_2\end{array}\right]\right)= ab\left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
- \(1\left[\begin{array}{c}x_1\\x_2\end{array}\right]= \left[\begin{array}{c}x_1\\x_2\end{array}\right]\text{.}\)
- There exists some \(\unknown\) such that \(\unknown\left[\begin{array}{c}x_1\\x_2\end{array}\right]= \left[\begin{array}{c}y_1\\y_2\end{array}\right]\text{.}\)
- \(a(\vec u+\vec v)=a\vec u+a\vec v\text{.}\)
- \((a+b)\vec v=a\vec v+b\vec v\text{.}\)
Fact 20.93.
Every Euclidean vector space \(\mathbb R^n\) satisfies the following properties, where \(\vec u,\vec v,\vec w\) are Euclidean vectors and \(a,b\) are scalars.
- Vector addition is associative: \(\vec u + (\vec v + \vec w)= (\vec u + \vec v) + \vec w\text{.}\)
- Vector addition is commutative: \(\vec u + \vec v= \vec v + \vec u\text{.}\)
- An additive identity exists: There exists some \(\vec z\) where \(\vec v + \vec z=\vec v\text{.}\)
- Additive inverses exist: There exists some \(-\vec v\) where \(\vec v + (-\vec v)=\vec z\text{.}\)
- Scalar multiplication is associative: \(a (b \vec v)=(ab) \vec v\text{.}\)
- 1 is a multiplicative identity: \(1 \vec v=\vec v\text{.}\)
- Scalar multiplication distributes over vector addition: \(a (\vec u + \vec v)=(a \vec u) + (a \vec v)\text{.}\)
- Scalar multiplication distributes over scalar addition: \((a+ b) \vec v=(a \vec v) + (b \vec v)\text{.}\)
Definition 20.94.
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 20.95.
Consider the set \(\IC\) of complex numbers with the usual definition for addition: \((a+b\mathbf i)\oplus(c+d\mathbf i)=(a+c)+(b+d)\mathbf i\text{.}\)
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\oplus(\vec v \oplus \vec w)
&=
(a+b\mathbf{i})\oplus((c+d\mathbf{i})\oplus(e+f\mathbf{i}))\\
&=
(a+b\mathbf{i})\oplus((c+e)+(d+f)\mathbf{i})\\
&=(a+c+e)+(b+d+f)\mathbf{i}
\end{align*}
\begin{align*}
(\vec u\oplus\vec v) \oplus \vec w
&=
((a+b\mathbf{i})\oplus(c+d\mathbf{i}))\oplus(e+f\mathbf{i})\\
&=((a+c)+(b+d)\mathbf{i})\oplus(e+f\mathbf{i})\\
&=(a+c+e)+(b+d+f)\mathbf{i}
\end{align*}
This proves that complex addition is associative: \(\vec u\oplus(\vec v \oplus \vec w) = (\vec u\oplus\vec v) \oplus \vec w\text{.}\) The seven other vector space properties may also be verified, so \(\IC\) is an example of a vector space.
Remark 20.96.
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 20.97.
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 20.98.
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 20.99.
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 20.100.
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)
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 \((x,y)\in V\text{.}\)
That is, pick appropriate values for \(\vec{z}=(\unknown,\unknown)\) and then simplify \((x,y)\oplus(\unknown,\unknown)\) into just \((x,y)\text{.}\)
(c)
Is \(V\) a vector space?
- Yes
- No
- More work is required
Remark 20.101.
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*}
satisfies all eight properties from Definition 20.94.
Thus, \(V\) is a vector space.
Activity 20.102.
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 \((0,-1)\oplus(z,w)\not=(0,-1)\) no matter what the values of \(z,w\) are.
(c)
Is \(V\) a vector space?
- Yes
- No
- More work is required
Activity 20.103.
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?
- Yes
- No
- More work is required
Subsection 20.5.3 Cooldown
Activity 20.104.
(a)
What are some objects that are important to you personally, academically, or otherwise that appear vector-like to you? What makes them feel vector-like? Which axiom for vector spaces does not hold for these objects, if any?
(b)
Our vector space axioms have eight properties. While these eight properties are enough to capture vectors, the objects that we study in the real world often have additional structures not captured by these axioms. What are some structures that you have encountered in other classes, or in previous experiences, that are not captured by these eight axioms?
Subsection 20.5.4 Videos
Subsection 20.5.5 Exercises
Exercises available at
https://tbil.org/preview/linear-algebra/exercises/#/bank/AT5/.Subsection 20.5.6 Mathematical Writing Explorations
Exploration 20.105.
- 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 20.106.
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 20.6
Subsection 20.5.7 Sample Problem and Solution
Sample problem Example C.16.
