How to find a basis for a tricky 2x 2 matrices vector space

Question

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this vector space. I need to compute the dimension of this vector space so how can i find a basis for this vector space??.

The issue finding a basis is basically i: how can i generate the real number entry "b" ?. This is an module theory problem so arguments that goes further than linear algebra are OK. Thanks and sorry for the sloppy gramar :(

Answer 1

You've probably already realized that your basis will include the two vectors $$ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\quad \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $$ so our problem is to find a $\Bbb Q$-basis for the space of all matrices of the form $$ \begin{bmatrix} 0 & r\\ 0 & 0 \end{bmatrix} $$ where $r\in\Bbb R$. Of course, this is equivalent to finding a basis for $\Bbb R$ as a vector space over $\Bbb Q$. It turns out that $\Bbb R$ is infinite dimensional when considered as a vector space over $\Bbb Q$, which might be part of your problem here. There is a lot written about this. Personally I find this blog post a nice place to read about this space.

Answer 2

Thanks bro, here is what i got : I arrive to the fact that in order to find a Q -basis for
the space of all matrices of the form : \begin{bmatrix} 0 & r\\ 0 & 0 \end{bmatrix} with r a real number . I can consider the Hamel Basis to generate r , thus a Q-basis for this vector space would be : B^ =\begin{bmatrix} 0 & B\\ 0 & 0 \end{bmatrix} where B is the hamel basis of reals numbers over rationals. And since reals over rationals as a vector space is an infinite dimension space with cardinality equal to aleph 1 then the cardinality of B is aleph 1 too this way the cardininality of B^ would be aleph 1 so the cardinality of B^^ plus the vectors : \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} is going to be the dimensión of the original Q - vector space which is aleph-1 +2 =aleph 1 :)