Basis?

[lollipop2046]

I have to show that for 2 distinct vectors u and v of a vector space V, for which {u, v} is a basis for V and a and b are nonzero scalars, then {u+v, au} is also basis for V.

do i show that they are lienarly independent and that i can convert it back to the form of {u, v}?

 

[Matt]

Just show that {u+v, au} is linearly independent. Then you are done.

 

[lollipop2046]

Thanks for replying..

so, just linearly dependency would be sufficient? Can you please tell me wHat is the theory behind? I thought in order for it to be a basis, it has to be independent and spans V....

 

[Matt]

Hi lollipop,

If the number of linearly independent vectors you have matches the dimension of the vector space, then those vectors form a basis for the vector space. You should have a theorem which says this.

In the case of your problem, the dimension of V is known to be 2. Therefore, if you can find 2 linearly independent vectors, then those 2 vectors automatically form a basis for V.

 

[lollipop2046]

Thanks a lot! I got it now!