2.7

Let A_{m×n} = ( \begin{matrix}r_{1} \\ r_{2} \\ ... \\ r_{m} \end{matrix} ) = ( c_{1} c_{2} ... c_{n})

Rank of A = dim(span{r_{1}, ..., r_{m}}) = dim(span{c_{1}, ..., c_{m}}) = maximum order of any nonvanishing minor determinant.

Ex: ( \begin{matrix}1 & 2 & 3 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 5 & 6 \end{matrix} ) ∼ ( \begin{matrix}1 & 3 & 2 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 5 & 0 & 6 \end{matrix} )

Let F : U ⊂ R^{n} → R^{m} ∈ C^{1}. Rank of F at x = rank of DF (x) F has rank k if F has rank k at each x. Det : M^{n×m} → R is a continuous function. Suppose rank DF (a) = k implies there exists V open such that a ∈ V and DF (x) ≥ k for all x ∈ V Ex: F (x_{1}, x_{2}) = (x_{1}x_{2} + 5, x_{1} + x_{2} − 3)

DF = ( \begin{matrix}x_{2} & x_{1} \\ 1 & 1 \end{matrix} )

Rank Theorem: Suppose A_{0} ⊂ R^{n}, B_{0} ⊂ R^{m}, F : A_{0} → B_{0} ∈ C^{1} a ∈ A_{0}, b ∈ B_{0}. Suppose rank F = k.

Then there exists A^{open} ⊂ A_{0} such that a ∈ A and B^{open} ⊂ B_{0} such that b ∈ B and G, H, C^{r} diffeomorphisms

such that G : A → U^{open} ⊂ R^{n}, H : B → V^{open} ⊂ R^{m} and

H ∘ F ∘ G^{−1}(x_{1}, ..., x_{n}) = (x_{1}, ..., x_{k}, 0, ..., 0)