Register to reply

Some vector space proofs

by Bertrandkis
Tags: proofs, space, vector
Share this thread:
Bertrandkis
#1
Jan15-08, 09:45 AM
P: 25
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].
Phys.Org News Partner Science news on Phys.org
Fungus deadly to AIDS patients found to grow on trees
Canola genome sequence reveals evolutionary 'love triangle'
Scientists uncover clues to role of magnetism in iron-based superconductors
HallsofIvy
#2
Jan15-08, 10:00 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,502
Quote Quote by Bertrandkis View Post
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
Yes, that's looks good. And you understand, I assume, that "u orthogonal to v1, v2 ..., vn" means u is orthogonal to each of v1, v2, ..., vn- that's where you get (u.v1)+ (u.v2)+ ...+ (u.vn)= 0+ 0+ ...+ 0= 0.

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]
I don't understand this. Why is [tex] Av1={\lambda}1x1[/tex]? Are you assuming each of the basis vectors is an eigenvector of A? That is not given in the hypothesis.

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].
Even if it were true that the original basis consists of eigenvectors of A, what you have done is show that Av1, Av2, ..., Avn span the space. You have not shown that they are independent. Also, you have not used the fact that A is nonsingular.

Better, I think, would be to use "proof by contradiction". Suppose Av1, Av2, ..., Avn were NOT independent. What would that tell you about v1, v2, ..., vn (remember that since A in nonsingular, it has an inverse matrix). Suppose Av1, Av2, ..., Avn does NOT span the space. That is, suppose there were some w such that a1Av1+ a2Av2+ ...+ anAvn was NOT equal to w for any choice of a1, a2, ..., an. What does that tell you about v1, v2, ..., vn and A-1w?


Register to reply

Related Discussions
Vector Proofs Precalculus Mathematics Homework 7
Vector Proofs Calculus & Beyond Homework 2
Vector proofs Precalculus Mathematics Homework 14
Vector Proofs in geometry General Math 1
Vector Proofs using vector components Introductory Physics Homework 2