Let u and v be distinct vectors of a vector space V. Show that if...

Kathleen Rausch

Kathleen Rausch

Answered

2022-01-07

Let u and v be distinct vectors of a vector space V. Show that if {u, v} is a basis for V and a and b are nonzero scalars, then both {u+v, au} and {au, bv} are also bases for V.

Answer & Explanation

Orlando Paz

Orlando Paz

Expert

2022-01-08Added 42 answers

Given, u and v be distinct vectors of a vector space V and {u, v} is a basis for V and a and b are nonzero scalars.
We have to show: both {u + v, au} and {au, bv} are also bases for V.
uv,a,b0
{u,v} is a basis for V {u+v,au} and {au,bv} are bases for V
From basis {u,v} we get dimension of V
{u,v} is a basis for V V is a 2-dimensional vector space and ()αu+βv=0α=β=0
Show {u+v,au} is a basis for V:
{u+v,au}:
A(u+v)+B(au)=0Au+Av+Bau=0
(A+Ba)u+(B)v=0
By the step (in second step)
A+Ba=0,a0
B=0
A=B=0{u+v,au} is linearly independent set of two vectors
{u+v,au} is a basis for V
(Proved)
xandir307dc

xandir307dc

Expert

2022-01-09Added 35 answers

here is the continuation of the solution:
Show {au,bv} is a basis for V
{au,bv}:
A(au)+B(bv)=0(Aa)u+(bb)v=0
By the step (in second step)
Aa=0,a0
Bb=0,b0
A=B=0{au,bv}: is linearly independent set of two vectors
{au,bv}: is a basis for V
(Proved)

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