Vector Spaces (V1)

Section 2.1 Vector Spaces (V1)

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 2.1.1.

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 scalars \(a,b\in\IR\text{,}\) 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 scalar \(c\) 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 2.1.2.

A vector space \(V\) is any collection of mathematical objects with associated addition \(\oplus\) and scalar multiplication \(\odot\) operations that satisfy the following properties. Let \(\vec u,\vec v,\vec w\) belong to \(V\text{,}\) and let \(a,b\) be scalar numbers.

\(\vec u\oplus (\vec v\oplus \vec w)= (\vec u\oplus \vec v)\oplus \vec w\text{.}\)
\(\vec u\oplus \vec v= \vec v\oplus \vec u\text{.}\)
There exists some \(\vec z\) where \(\vec v\oplus \vec z=\vec v\text{.}\)
There exists some \(-\vec v\) where \(\vec v\oplus (-\vec v)=\vec z\text{.}\)
\(a(b\odot\vec v)=(ab)\odot\vec v\text{.}\)
\(1\odot\vec v=\vec v\text{.}\)
\(a\odot(\vec u\oplus \vec v)=(a\odot\vec u)\oplus(a\odot\vec v)\text{.}\)
\((a+ b)\odot\vec v=(a\odot\vec v)\oplus(b\odot \vec v)\text{.}\)

Observation 2.1.3.

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 2.1.4.

The space of \(m \times n\) matrices

\begin{equation*} M_{m,n}=\setBuilder{\left[\begin{array}{c}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 2.1.5.

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 2.1.6.

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 2.1.2.

Consider the set \(V=\setBuilder{(x,y)}{y=e^x}\) with operations 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_1,y_1)=(cx_1,y_1^c)\text{.} \end{equation*}

(a)

Show that \(V\) satisfies the distributive property

\begin{equation*} (a+b)\odot (x_1,y_1)=\left(a\odot (x_1,y_1)\right)\oplus \left(b\odot (x_1,y_1)\right) \end{equation*}

by simplifying both sides and verifying they are the same expression.

(b)

Show that \(V\) contains an additive identity element satisfying

\begin{equation*} (x_1,y_1)\oplus\vec{z}=(x_1,y_1) \end{equation*}

for all \((x_1,y_1)\in V\) by choosing appropriate values for \(\vec{z}=(\unknown,\unknown)\text{.}\)

Remark 2.1.7.

It turns out \(V=\setBuilder{(x,y)}{y=e^x}\) with operations 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_1,y_1)=(cx_1,y_1^c) \end{equation*}

satisifes all eight properties.

Thus, \(V\) is a vector space.

Activity 2.1.3.

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) \hspace{3em} c\odot (x_1,y_1)=(x_1^c,y_1+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 by showing that \((0,-1)\oplus\vec z\not=(0,-1)\) no matter how \(\vec z=(z,w)\) is chosen.

(c)

Is \(V\) a vector space?

Activity 2.1.4.

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_1,y_1)=(cx_1,cy_1) . \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?

Exercises 2.1.1 Exercises

1.

Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= \left(4 \, x_{1} + x_{2},\,y_{1} + 3 \, y_{2}\right) \\ c \odot (x,y) &= \left(c x,\,c y\right) . \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

\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*}

(b) Explain why \(V\) nonetheless is not a vector space.

Answer

\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

vector addition is not associative
vector addition is not commutative
scalar multiplication does not distribute over scalar addition

2.

Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\ c \odot (x,y) &= \left(c^{2} x,\,c^{3} y\right) . \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

\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*}

(b) Explain why \(V\) nonetheless is not a vector space.

Answer

\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

scalar multiplication does not distribute over scalar addition

3.

Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= \left(x_{1} + x_{2},\,y_{1} + y_{2}\right) \\ c \odot (x,y) &= \left(c x,\,c y - 3 \, c + 3\right) . \end{align*}

(a) Show that scalar multiplication is associative, that is:

\begin{equation*} a\odot(b\odot (x,y))=(ab)\odot(x,y). \end{equation*}

(b) Explain why \(V\) nonetheless is not a vector space.

Answer

\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

scalar multiplication does not distribute over vector addition
scalar multiplication does not distribute over scalar addition

4.

Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= \left(x_{1} + x_{2},\,y_{1} + y_{2} - 5\right) \\ c \odot (x,y) &= \left(c x,\,c y\right) . \end{align*}

(a) Show that vector addition is associative, that is:

\begin{equation*} \left((x_1,y_1)\oplus(x_2,y_2)\right)\oplus(x_3,y_3)=(x_1,y_1)\oplus\left((x_2,y_2)\oplus(x_3,y_3)\right). \end{equation*}

(b) Explain why \(V\) nonetheless is not a vector space.

Answer

\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

scalar multiplication does not distribute over vector addition
scalar multiplication does not distribute over scalar addition

5.

Let \(V\) be the set of all pairs \((x,y)\) of real numbers together with the following operations:

\begin{align*} (x_1,y_1)\oplus (x_2,y_2)&= \left(4 \, x_{1} + 4 \, x_{2},\,4 \, y_{1} + 4 \, y_{2}\right) \\ c \odot (x,y) &= \left(c x,\,c y\right) . \end{align*}

(a) Show that scalar multiplication distributes over vector addition, that is:

\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*}

(b) Explain why \(V\) nonetheless is not a vector space.

Answer

\(V\) is not a vector space, which may be shown by demonstrating that any one of the following properties do not hold:

vector addition is not associative
scalar multiplication does not distribute over scalar addition

Additional exercises available at checkit.clontz.org.