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I have these two sets (I couldn't use the notation {} in latex, don't know how).

V is the set of matrices spanned by these 3 matrices written below. W is a set of 2x3 matrices applying the rule a+e=c+f

\[V=span(\begin{pmatrix} 1 &1 &1 \\ 1 &3 &7 \end{pmatrix},\begin{pmatrix} 0 &0 &0 \\ 1 &1 &1 \end{pmatrix},\begin{pmatrix} 0 &0 &0 \\ 0 &2 &3 \end{pmatrix})\]

\[W=(\begin{pmatrix} a &b &c \\ e &f &g \end{pmatrix}|a+e=c+f)\]

For both sets, I need to determine if it is a sub-vector space, I need to find the basis and dimension.

For W, I think it is fairly easy to prove it is a vector space (sub space - assuming R2x3 is a vector space, which I am allowed to do in this example). However I am not so sure about the dimension. Can I write a=c+f-e and say that dim(W)=5 ?

My bigger problem is V. I managed to show that a matrix in V is from the shape:

\[\begin{pmatrix} a &a &a \\ a+b &3a+b+2c &7a+b+3c \end{pmatrix}\]

but I am not sure how from here I show that it is a vector space (I am looking for the 3 rules: 0 matrix, + and scalar *). In addition, I don't know what the dimension is (my intuition say dim(V)=3)

Can you assist please ?

Thank you in advance !