# Math Help - Bases and Dimension....

1. ## Bases and Dimension....

Hello can you please help me with this bases and dimension based problems:

Q1) Give three different bases for F2 and for M[2x2] (F).

Q2) 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.

Thanks a lot for the help!

2. Originally Posted by Vedicmaths
Hello can you please help me with this bases and dimension based problems:

Q2) 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.

Thanks a lot for the help!
1. it should be easy to show that $\{u+v, au\}$ is a linearly independent set. as for the other criterion, it is enough to show that you can express $u$ and $v$ in terms of $u+v$ and $au$: $u = 0(u+v)+ a^{-1}au$ and $v = 1(u+v) - a^{-1}au$.

you can do similar arguments for the other set..