06-240/Classnotes For Tuesday September 26

Vector Spaces

Example 5.

${\displaystyle {\mbox{Polynomials:}}{}_{}^{}}$

${\displaystyle 7x^{3}+9x^{2}-2x+\pi \ }$

${\displaystyle {\mbox{Let }}{\mathcal {F}}\ {\mbox{be a field.}}}$

${\displaystyle P({\mathcal {F}})={\bigg \{}\sum _{i=1}^{n}a_{i}x^{i}:n\in \mathbb {Z} \,\ n\geq 0\ \forall i\ \ a_{i}\in {\mathcal {F}}{\bigg \}}{}_{}^{}}$

${\displaystyle {\mbox{Addition of polynomials is defined in the expected way:}}{}_{}^{}}$

${\displaystyle \sum _{i=0}^{n}a_{i}x^{i}+\sum _{i=1}^{m}b_{i}x^{i}=\sum _{i=0}^{max(m,n)}{(a_{i}+b_{i})}x^{i}}$

Theorem 1.(Cancellation law for vector spaces)

${\displaystyle {\mbox{If in a vector space x+z=y+z then x=y.}}{}_{}^{}}$

Proof:

${\displaystyle {\mbox{Add w to both sides of a given equation where w is an element}}{}_{}^{}}$ ${\displaystyle {\mbox{for which z+w=0 (exists by VS4)}}{}_{}^{}}$

${\displaystyle (x+y)+w=(y+z)+w\ }$

${\displaystyle x+(z+w)=y+(z+w)\ {\mbox{(by VS2)}}{}_{}^{}}$

${\displaystyle x+0=y+0\ {\mbox{(by the choice of w)}}{}_{}^{}}$

${\displaystyle x=y\ {\mbox{(by VS3)}}{}_{}^{}}$

Theorem 2. "0 is unique"

${\displaystyle {\mbox{If some z}}\in {\mbox{V satisfies x+z=0 for some x}}\in {\mbox{V then z=0.}}{}_{}^{}}$

Proof:"

${\displaystyle x+z=x+0\ }$

${\displaystyle z+x=0+x\ }$
${\displaystyle z=0\ }$

Theorem 3. "negatives are unique"

${\displaystyle {\mbox{If x+y=0 and x+z=0 then y=z.}}{}_{}^{}}$

Theorem 4.

a)${\displaystyle 0_{F}.x=0_{V}\ }$

b)${\displaystyle a.0_{V}=0_{V}\ }$

c)${\displaystyle (-a)x=a(-x)=-(ax)\ }$

Theorem 5.

${\displaystyle {\mbox{If }}x_{i}\ {\mbox{ i=1,...,n are in V then }}\sum {x_{i}}=x_{1}+x_{2}+...+x_{n}\ {\mbox{ makes sense whichever way you parse it.}}{}_{}^{}}$

${\displaystyle {\mbox{(From VS1 and VS2)}}{}_{}^{}}$

Subspaces

Definition

${\displaystyle {\mbox{Let V be a vector space. A subspace of V is a subset W of V which is a vector space in itself under the operations is inherits from V.}}{}_{}^{}}$

Theorem

${\displaystyle W\subset V\ {\mbox{is a subspace of V iff}}{}_{}^{}}$

1. ${\displaystyle \forall x,y\in W\ \ x+y\in W\ }$
2. ${\displaystyle \forall a\in F,\ \forall x\in W\ \ ax\in W\ }$
3. ${\displaystyle 0\in W\ }$

Proof
${\displaystyle \Rightarrow }$

${\displaystyle {\mbox{Assume W is a subspace. If x,y}}\in {\mbox{W then x+y}}\in {\mbox{W because W is a vector space in itself. Likewise for a.w.}}{}_{}^{}}$

${\displaystyle \Leftarrow }$

${\displaystyle {\mbox{Assume W}}\subset {\mbox{V for which }}x,y\in W\Rightarrow x+y\in W\ ;x\in W,a\in F\Rightarrow ax\in W.\ {}_{}^{}}$

${\displaystyle {\mbox{We need to show that W is a vector space. Addition and multiplication are clearly defined on W so we just need to check VS1-VS8.}}{}_{}^{}}$

${\displaystyle {\mbox{Indeed, VS1, VS2, VS5, VS6, VS7, and VS8 hold in V hence in W.}}{}_{}^{}}$

${\displaystyle {\mbox{VS3-pick any x}}\in W\ \ 0=0.x\in W\ by\ 2.\ {}_{}^{}}$

${\displaystyle {\mbox{VS4-given x in W, take y=(-1).x}}\in W\ and\ x+y=0.\ {}_{}^{}}$

Examples

Example 1.

Definition

${\displaystyle {\mbox{If A}}\in M_{m\times n}(F){\mbox{ the transpose of A, }}A^{t}{\mbox{ is the matrix }}(A^{t})_{ij}:=A_{ji}.{}_{}^{}}$

${\displaystyle {\begin{pmatrix}2&3&\pi \ \\7&8&-2\end{pmatrix}}^{t}={\begin{pmatrix}2&7\\3&8\\\pi \ &-2\end{pmatrix}}}$

${\displaystyle {\mbox{Then:}}{}_{}^{}}$

1. ${\displaystyle A^{t}\in M_{n\times m}(F)\ }$
2. ${\displaystyle (A^{t})^{t}=A\ }$
3. ${\displaystyle (A+B)^{t}=A^{t}+B^{t}\ }$
4. ${\displaystyle (cA)^{t}=c(A^{t})\ \forall c\in F\ }$

Definition

${\displaystyle A\in M_{n\times n}(F){\mbox{ is called symmetric if }}A^{t}=A.\ {}_{}^{}}$

Claim

${\displaystyle V=M_{n\times n}(F)\ {\mbox{ is a vector space. Let }}\ W={\big \{}{\mbox{symmetric A-s in V}}{\big \}}={\big \{}A\in V:A^{t}=A{\big \}}\ {\mbox{ then W is a subspace of V.}}{}_{}^{}}$

Proof

1.${\displaystyle {\mbox{Need to show that if }}A\in Wand\ B\in W\ then\ A+B\in W.\ {}_{}^{}}$

${\displaystyle A^{t}=A,\ B^{t}=B\ }$

${\displaystyle (A+B)^{t}=A^{t}+B^{t}=A+B\ so\ A+B\in W.}$

${\displaystyle {\mbox{If }}A\in W,\ c\in F{\mbox{ need to show }}cA\in W{}_{}^{}}$

${\displaystyle (cA)^{t}=cA^{t}=cA\ \Rightarrow cA\in W}$

3.${\displaystyle 0_{M}={\begin{pmatrix}0&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &0\end{pmatrix}}\Rightarrow 0^{t}=0\ so\ 0\in W}$

Example 2.

${\displaystyle V=M_{n\times n}(F)}$

${\displaystyle A=A_{ij}\ \ trA=\sum _{i=1}^{n}A_{ii}\ {\mbox{(the trace of A)}}}$

${\displaystyle {\mbox{Properties of tr:}}{}_{}^{}}$

1. ${\displaystyle tr0_{M}=0\ }$
2. ${\displaystyle tr(A+B)=tr(A)+tr(B)\ }$
3. ${\displaystyle tr(cA)=c.trA\ }$

${\displaystyle A={\begin{pmatrix}1&0\\0&0\end{pmatrix}}\ \ B={\begin{pmatrix}0&0\\0&1\end{pmatrix}}\ }$
${\displaystyle trA=1\ \ trB=1\ }$

${\displaystyle Set\ \ W={\big \{}A\in V:trA=0{\big \}}={\bigg \{}{\begin{pmatrix}1&7\\\pi \ &-1\end{pmatrix}},...{\bigg \}}\ }$

Claim

${\displaystyle {\mbox{W is a subspace.}}{}_{}^{}}$

${\displaystyle {\mbox{Indeed,}}{}_{}^{}}$

1. ${\displaystyle A,B\in W\Rightarrow trA=0=trB\ \ tr(A+B)=tr(A)+tr(B)=0+0=0\ so\ A+B\in W\ }$
2. ${\displaystyle A\in W\ \ trA=0\ \ tr(cA)=c(trA)=c.0=0\ so\ cA\in W\ }$
3. ${\displaystyle tr0_{M}=0\ \ 0_{M}\in W\ }$

Example 3.

${\displaystyle W_{3}={\big \{}A\in M_{n\times n}(F):trA=1{\big \}}{\mbox{ Not a subspace.}}{}_{}^{}}$
${\displaystyle A,B\in W_{3}\Rightarrow tr(A+B)=trA+trB=1+1=2\ so\ A+B\ \not \in W_{3}\ }$

Theorem
${\displaystyle {\mbox{The intersection of two subspaces of the same space is always a subspace.}}{}_{}^{}}$
${\displaystyle {\mbox{Assume }}W_{1}\subset V{\mbox{ is a subspace of V, }}W_{2}\subset V{\mbox{ is a subspace of V, then }}W_{1}\cap W_{2}={\big \{}x:x\in W_{1}\ and\ x\in W_{2}{\big \}}{\mbox{ is a subspace.}}{}_{}^{}}$
${\displaystyle {\mbox{However, }}W_{1}\cup W_{2}={\big \{}x:x\in W_{1}\ or\ W_{2}{\big \}}{\mbox{ is most often not a subspace.}}{}_{}^{}}$

Proof

1.${\displaystyle {\mbox{Assume }}x,y\in W_{1}\cap W_{2}{\mbox{ , that is, }}x\in W_{1},x\in W_{2},y\in W_{1},y\in W_{2}.\ {}_{}^{}}$
${\displaystyle x+y\in W_{1}\ as\ x,y\in W_{1}{\mbox{ and }}W_{1}{\mbox{ is a subspace}}{}_{}^{}}$
${\displaystyle x+y\in W_{2}\ as\ x,y\in W_{2}{\mbox{ and }}W_{2}{\mbox{ is a subspace}}{}_{}^{}}$
${\displaystyle {\mbox{So }}x+y\in W_{1}\cap W_{2}.\ {}_{}^{}}$

2.${\displaystyle {\mbox{If}}\ x\in W_{1}\cap W_{2}\ then\ x\in W_{1}\Rightarrow cx\in W_{1}\ ,\ x\in W_{2}\Rightarrow cx\in W_{2}\ \Rightarrow cx\in W_{1}\cap W_{2}.\ {}_{}^{}}$

3.${\displaystyle 0\in W_{1}\ ,\ 0\in W_{2}\Rightarrow 0\in W_{1}\cap W_{2}.\ }$