Introduction to Vector Spaces

A vector space (over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} ) consists of a set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} along with two operations "$+$" and "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot} " subject to these conditions.

1. For any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v,\vec w\in V:\vec v+\vec w\in V} .
2. For any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v,\vec w\in V:\vec v+\vec w=\vec w+\vec v} .
3. For any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec u,\vec v,\vec w\in V:(\vec v+\vec w)+\vec u=\vec v+(\vec w+\vec u)} .
4. There is a zero vector ${\vec {0}}\in V$ such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v+\vec0=\vec v} for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v\in V} .
5. Each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v\in V} has an additive inverse Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec w\in V} such that ${\vec {w}}+{\vec {v}}={\vec {0}}$.
6. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is a scalar, that is, a member of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v\in V} then the scalar multiple Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\cdot\vec v } is in $V$.
7. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r,s\in\R} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v\in V} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r+s)\cdot\vec v=r\cdot\vec v+s\cdot\vec v} .
8. If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\in\R} and ${\vec {v}},{\vec {w}}\in V$ , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\cdot(\vec v+\vec w)=r\cdot\vec v+r\cdot\vec w} .
9. If $r,s\in \mathbb {R}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v\in V} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (rs)\cdot\vec v=r\cdot(s\cdot\vec v)} .
10. For any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v\in V} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\cdot\vec v=\vec v} .

Remark: Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. For instance, in condition 7 "$(r+s)\cdot {\vec {v}}=r\cdot {\vec {v}}+s\cdot {\vec {v}}$", the first "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +} " is the real number addition operator while the "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +} " to the right of the equals sign represents vector addition in the structure Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . These expressions aren't ambiguous because, e.g., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} and $s$ are real numbers so "$r+s$" can only mean real number addition.

Lemma 1.17: In any vector space Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } , for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}\in V } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\in\mathbb{R} } , we have

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\cdot\vec{v}=\vec{0} } , and
2. $(-1\cdot {\vec {v}})+{\vec {v}}={\vec {0}}$, and
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\cdot\vec{0}=\vec{0} } .

Proof: For 1, note that ${\vec {v}}=(1+0)\cdot {\vec {v}}={\vec {v}}+(0\cdot {\vec {v}})$. Add to both sides the additive inverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} } , the vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w} } such that ${\vec {w}}+{\vec {v}}={\vec {0}}$.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rl} \vec{w}+\vec{v} &=\vec{w}+\vec{v}+0\cdot\vec{v} \\ \vec{0} &=\vec{0}+0\cdot\vec{v} \\ \vec{0} &=0\cdot\vec{v} \end{array}}

The second item is easy: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1\cdot\vec{v})+\vec{v}=(-1+1)\cdot\vec{v}=0\cdot\vec{v}=\vec{0} } shows that we can write "$-{\vec {v}}\,$" for the additive inverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v} } without worrying about possible confusion with $(-1)\cdot {\vec {v}}$.

For 3, this $r\cdot {\vec {0}}=r\cdot (0\cdot {\vec {0}})=(r\cdot 0)\cdot {\vec {0}}={\vec {0}}$ will do.

Example 1

The set $\mathbb {R} ^{2}$ is a vector space if the operations "$+$" and "$\cdot$" have their usual meaning.

${\binom {x_{1}}{x_{2}}}+{\binom {y_{1}}{y_{2}}}={\binom {x_{1}+y_{1}}{x_{2}+y_{2}}}\qquad r\cdot {\binom {x_{1}}{x_{2}}}={\binom {rx_{1}}{rx_{2}}}$

We shall check all of the conditions.

There are five conditions in item 1. For 1, closure of addition, note that for any $v_{1},v_{2},w_{1},w_{2}\in \mathbb {R}$ the result of the sum

${\binom {v_{1}}{v_{2}}}+{\binom {w_{1}}{w_{2}}}={\binom {v_{1}+w_{1}}{v_{2}+w_{2}}}$

is a column array with two real entries, and so is in $\mathbb {R} ^{2}$ . For 2, that addition of vectors commutes, take all entries to be real numbers and compute

${\binom {v_{1}}{v_{2}}}+{\binom {w_{1}}{w_{2}}}={\binom {v_{1}+w_{1}}{v_{2}+w_{2}}}={\binom {w_{1}+v_{1}}{w_{2}+v_{2}}}={\binom {w_{1}}{w_{2}}}+{\binom {v_{1}}{v_{2}}}$

(the second equality follows from the fact that the components of the vectors are real numbers, and the addition of real numbers is commutative). Condition 3, associativity of vector addition, is similar.

${\begin{aligned}{\Bigg (}{\binom {v_{1}}{v_{2}}}+{\binom {w_{1}}{w_{2}}}{\Bigg )}+{\binom {u_{1}}{u_{2}}}&={\binom {(v_{1}+w_{1})+u_{1}}{(v_{2}+w_{2})+u_{2}}}\\&={\binom {v_{1}+(w_{1}+u_{1})}{v_{2}+(w_{2}+u_{2})}}\\&={\binom {v_{1}}{v_{2}}}+{\Bigg (}{\binom {w_{1}}{w_{2}}}+{\binom {u_{1}}{u_{2}}}{\Bigg )}\end{aligned}}$

For the fourth condition we must produce a zero element — the vector of zeroes is it.

${\binom {v_{1}}{v_{2}}}+{\binom {0}{0}}={\binom {v_{1}}{v_{2}}}$

For 5, to produce an additive inverse, note that for any $v_{1},v_{2}\in \mathbb {R}$ we have

${\binom {-v_{1}}{-v_{2}}}+{\binom {v_{1}}{v_{2}}}={\binom {0}{0}}$

so the first vector is the desired additive inverse of the second.

The checks for the five conditions having to do with scalar multiplication are just as routine. For 6, closure under scalar multiplication, where $r,v_{1},v_{2}\in \mathbb {R}$ ,

$r\cdot {\binom {v_{1}}{v_{2}}}={\binom {rv_{1}}{rv_{2}}}$

is a column array with two real entries, and so is in $\mathbb {R} ^{2}$ . Next, this checks 7.

$(r+s)\cdot {\binom {v_{1}}{v_{2}}}={\binom {(r+s)v_{1}}{(r+s)v_{2}}}={\binom {rv_{1}+sv_{1}}{rv_{2}+sv_{2}}}=r\cdot {\binom {v_{1}}{v_{2}}}+s\cdot {\binom {v_{1}}{v_{2}}}$

For 8, that scalar multiplication distributes from the left over vector addition, we have this.

$r\cdot {\Bigg (}{\binom {v_{1}}{v_{2}}}+{\binom {w_{1}}{w_{2}}}{\Bigg )}={\binom {r(v_{1}+w_{1})}{r(v_{2}+w_{2})}}={\binom {rv_{1}+rw_{1}}{rv_{2}+rw_{2}}}=r\cdot {\binom {v_{1}}{v_{2}}}+r\cdot {\binom {w_{1}}{w_{2}}}$

The ninth

$(rs)\cdot {\binom {v_{1}}{v_{2}}}={\binom {(rs)v_{1}}{(rs)v_{2}}}={\binom {r(sv_{1})}{r(sv_{2})}}=r\cdot \left(s\cdot {\binom {v_{1}}{v_{2}}}\right)$

and tenth conditions are also straightforward.

$1\cdot {\binom {v_{1}}{v_{2}}}={\binom {1v_{1}}{1v_{2}}}={\binom {v_{1}}{v_{2}}}$

In a similar way, each $\mathbb {R} ^{n}$ is a vector space with the usual operations of vector addition and scalar multiplication. (In $\mathbb {R} ^{1}$, we usually do not write the members as column vectors, i.e., we usually do not write "$(\pi )$". Instead we just write "$\pi$".)

Example 2

The set $F=\{a\cos \theta +b\sin \theta \,{\big |}\,a,b\in \mathbb {R} \}$ of real-valued functions of the real variable $\theta$ is a vector space under the operations

$(a_{1}\cos \theta +b_{1}\sin \theta )+(a_{2}\cos \theta +b_{2}\sin \theta )=(a_{1}+a_{2})\cos \theta +(b_{1}+b_{2})\sin \theta$

and

$r\cdot (a\cos \theta +b\sin \theta )=(ra)\cos \theta +(rb)\sin \theta$

inherited from the space in the prior example. (We can think of $F$ as "the same" as $\mathbb {R} ^{2}$ in that $a\cos \theta +b\sin \theta$ corresponds to the vector with components $a$ and $b$.)

Resources

• Definition and Examples of Vector Spaces, WikiBooks
• Linear Algebra: Vector Spaces and Subspaces, University of Houston