(adsbygoogle = window.adsbygoogle || []).push({}); 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].

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# Some vector space proofs

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