Finding Dimension of F intersection with G

[stone comet]

Would an appropriate approach be calculating the rank of the matrix, with columns being the 5 respective vectors?
Using mathematica, I computed it to be of rank 3. Thus the dimension of F intersection with G is 3?

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[stone comet]

The written question in the pic, asks for finding Dim(F+G), however I just wanna make sure I'm finding Dim(F intersection G ) correctly

[limpid tiger]

Well you already have your answer take a vectorial space E if you take two subspaces F and G then F+G={x in E such that there exist two vectors y and z in F and G such that x=y+z} so if you consider that F and G are both finite dimension then take (ei) a basis of F and (fi) one of G then for any x in F+G ( dim(F)=p dim(G)=k)x=y+z with y and z in F and G respectively so if y=y1e1+…+ypep and z=z1f1+…+zkfk then you have x=y1e1+…+ypep+z1f1+…+zkfk and if one or more of the elements of the basis of G is a linear combination of the basis of F then the basis of F+G is the basically the set of vectors that generates both basis so basically it has the same cardinal as the rank of both family of vectors combined

And here using Grassmans formula you’d see that the dimension of F inter G is actually the dimension of the kernel of your matrice

(Rank theorem)

To conclude here the rank of the matrice is the answer

[stone comet]

Much appreciated! @limpid tiger