Let u and v be distinct vectors of a vector space V. Show that if...
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
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.
is a basis for V and are bases for V
From basis we get dimension of V
is a basis for V V is a 2-dimensional vector space and
Show is a basis for V:
:
By the step (in second step)
is linearly independent set of two vectors
is a basis for V
(Proved)
xandir307dc
Expert
2022-01-09Added 35 answers
here is the continuation of the solution:
Show is a basis for V
By the step (in second step) is linearly independent set of two vectors is a basis for V
(Proved)