- #1
Bertrandkis
- 25
- 0
Question 1
Let u, v1,v2 ... vn be vectors in [tex]R^{n}[/tex]. Show that if u is orthogonal to v1,v2 ...vn then u is orthogonal to every vector in span{v1,v2...vn}
My attempt
if u is orthogonal to v1,v2 ...vn then[tex] (u.v1)+(u.v2)+...+(u.vn)=0[/tex]
Let w be a vector in span{v1,v2...vn} therefore
[tex] w=c1v1+c2v2+...+cnvn [/tex]
[tex] u.w=u(c1v1+c2v2+...+cnvn)[/tex]
=>[tex] c1(u.v1)+c2(u.v2)+...+cn(u.vn) =0 [/tex]
So u is orthogonal to w
Question 2
Let [tex] \{v1,v2...vn \}[/tex] be a basis for the n-dimensional vector space [tex]R^{n}[/tex].
Show that if A is a non singular matrix nxn then [tex] \{Av1,Av2...Avn \} [/tex] is also a basis for [tex]R^{n}[/tex].
Let w be a vector in [tex]R^{n}[/tex] therefore w can be written a linear combination of vectos in it's basis
[tex] x=c1v1+c2v2+...+cnvn [/tex]
[tex] Av1={\lambda}1x1[/tex],[tex] Av2={\lambda}2x2[/tex] ...[tex] Avn={\lambda}3xn[/tex]
so
[tex]Ax=A(c1v1+c2v2+...+cnvn) [/tex]
[tex]Ax={\lambda}1c1v1+{\lambda}2c2v2+...+{\lambda}ncnvn) [/tex]
therefore [tex] \{Av1,Av2...Avn \} [/tex] is also a basis for [tex]R^{n}[/tex].
Let u, v1,v2 ... vn be vectors in [tex]R^{n}[/tex]. Show that if u is orthogonal to v1,v2 ...vn then u is orthogonal to every vector in span{v1,v2...vn}
My attempt
if u is orthogonal to v1,v2 ...vn then[tex] (u.v1)+(u.v2)+...+(u.vn)=0[/tex]
Let w be a vector in span{v1,v2...vn} therefore
[tex] w=c1v1+c2v2+...+cnvn [/tex]
[tex] u.w=u(c1v1+c2v2+...+cnvn)[/tex]
=>[tex] c1(u.v1)+c2(u.v2)+...+cn(u.vn) =0 [/tex]
So u is orthogonal to w
Question 2
Let [tex] \{v1,v2...vn \}[/tex] be a basis for the n-dimensional vector space [tex]R^{n}[/tex].
Show that if A is a non singular matrix nxn then [tex] \{Av1,Av2...Avn \} [/tex] is also a basis for [tex]R^{n}[/tex].
Let w be a vector in [tex]R^{n}[/tex] therefore w can be written a linear combination of vectos in it's basis
[tex] x=c1v1+c2v2+...+cnvn [/tex]
[tex] Av1={\lambda}1x1[/tex],[tex] Av2={\lambda}2x2[/tex] ...[tex] Avn={\lambda}3xn[/tex]
so
[tex]Ax=A(c1v1+c2v2+...+cnvn) [/tex]
[tex]Ax={\lambda}1c1v1+{\lambda}2c2v2+...+{\lambda}ncnvn) [/tex]
therefore [tex] \{Av1,Av2...Avn \} [/tex] is also a basis for [tex]R^{n}[/tex].