What is the dimension of the graph of F?

[rtgt]

Hi everyone,

The problem that I'm having issues with reads:
"Let F: R^k →Rⁿ be a linear map. Recall that the graph G(F) of F is the subset of R^k × Rⁿ = R^k+n given by
G(F)={(x,y)∈R^k ×Rⁿ : y=F((x)}"
It first asked me to show that G(F) is a vector subspace of R^k+n which I did just by the definition of vector subspaces.

Then, though it asks for the dimension of G(F). How exactly do I go about finding that?

Any help is greatly appreciated.
Thanks!

 

[fresh_42]

We need to find $\dim \{\,(x,Fx)\,|\,x\in \mathbb{R}^k\,\}$. This is the image of the linear transformation $(1,F)\, : \,\mathbb{R}^k \longrightarrow \mathbb{R}^{k+n}\, , \,x\longmapsto (x,Fx)$. The dimension of this image is the rank of the linear transformation, hence the matrix rank of $\begin{bmatrix}\mathbb{1}_k \\ F\end{bmatrix}$. Now whatever $F$ does, we have the full $\mathbb{R}^k$ in the image and cannot get more.