06-240/Classnotes For Tuesday September 26

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Links to Classnotes

• Classnote for Tuesday Sept 26 [1]
• PDF file by User:Alla: Week 3 Lecture 1 notes

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Vector Spaces

Example 5.

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

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

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

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

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

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

Theorem 1.(Cancellation law for vector spaces)

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

Proof:

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

$(x+y)+w=(y+z)+w\$

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

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

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

Theorem 2. "0 is unique"

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

Proof:"

$x+z=x+0\$

$z+x=0+x\$
$z=0\$

Theorem 3. "negatives are unique"

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

Theorem 4.

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

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

c)$(-a)x=a(-x)=-(ax)\$

Theorem 5.

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

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

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Subspaces

Definition

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

Theorem

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

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

Proof
$\Rightarrow$

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

$\Leftarrow$

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

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

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

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

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

Examples

Example 1.

Definition

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

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

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

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

Definition

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

Claim

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

Proof

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

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

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

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

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

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

Example 2.

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

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

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

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

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

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

Claim

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

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

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

Example 3.

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

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

Proof

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

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

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