Let u and v be distinct vectors of a vector
spacev. show that if {u, v} is a basis for v anda
is a non-zero scalar, then {u+v, au} is also abasis for v.

See the proof below.

Step-by-step explanation:

For this proof we need to begin with the assumption, on this case the vectors u and v are different

We are assuming that {u,v} is a basis of V and distinct, so then we have that V is a 2 dimensional vector space and we have the following condition:

if    (1)

We need to show that {u+v,au} is also a basis for V, so then we need to show this:

That is equivalent to:

We can take common factor u and we got this:

from condition 1 we need to have this:

So then and

And we will see that so then we can conclude that {u+v, au} is a linearly independent set of two vectors and are a basis for V.

{u+v, au} is a basis for the space V.

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step-by-step explanation: