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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 2.1.2.

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 scalars \(a,b\in\IR\text{,}\) and invalid if it does not.

  1. \(\vec u+(\vec v+\vec w)= (\vec u+\vec v)+\vec w\text{.}\)

  2. \(\vec u+\vec v= \vec v+\vec u\text{.}\)

  3. There exists some \(\vec z\) where \(\vec v+\vec z=\vec v\text{.}\)

  4. There exists some \(-\vec v\) where \(\vec v+(-\vec v)=\vec z\text{.}\)

  5. 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 \(\vec v\)

  6. \(a(b\vec v)=(ab)\vec v\text{.}\)

  7. \(1\vec v=\vec v\text{.}\)

  8. If \(\vec u\not=\vec 0\text{,}\) then there exists some scalar \(c\) such that \(c\vec u=\vec v\text{.}\)

  9. \(a(\vec u+\vec v)=a\vec u+a\vec v\text{.}\)

  10. \((a+b)\vec v=a\vec v+b\vec v\text{.}\)

Definition 2.1.3.

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.

  1. Vector addition is associative: \(\vec u\oplus (\vec v\oplus \vec w)= (\vec u\oplus \vec v)\oplus \vec w\text{.}\)

  2. Vector addition is commutative: \(\vec u\oplus \vec v= \vec v\oplus \vec u\text{.}\)

  3. An additive identity exists: There exists some \(\vec z\) where \(\vec v\oplus \vec z=\vec v\text{.}\)

  4. Additive inverses exist: There exists some \(-\vec v\) where \(\vec v\oplus (-\vec v)=\vec z\text{.}\)

  5. Scalar multiplication is associative: \(a\odot(b\odot\vec v)=(ab)\odot\vec v\text{.}\)

  6. 1 is a multiplicative identity: \(1\odot\vec v=\vec v\text{.}\)

  7. Scalar multiplication distributes over vector addition: \(a\odot(\vec u\oplus \vec v)=(a\odot\vec u)\oplus(a\odot\vec v)\text{.}\)

  8. Scalar multiplication distributes over scalar addition: \((a+ b)\odot\vec v=(a\odot\vec v)\oplus(b\odot \vec v)\text{.}\)

Observation 2.1.4.

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 2.1.5.

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 2.1.6.

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 2.1.7.

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 2.1.8.

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 2.1.9.

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 2.1.10.

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 2.1.11.

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?

Subsection 2.1.1 Videos

Figure 2.1.12. Video: Verifying that a vector space property holds
Figure 2.1.13. Video: Showing something is not a vector space

Exercises 2.1.2 Exercises

Exercises available at checkit.clontz.org 1 .

https://checkit.clontz.org/#/banks/tbil-la/V1/