Prove by induction that A(1,n)=n+2 Use this to prove by induction that A(2,n)=2(n+3)-3 Use this to prove by induction that A(3,n)=2^{n+3}-3 And use this to prove by induction that A(4,n)=^{n+3}2-3 ...

The column space of A is a subspace of \Bbb R^n that is spanned by the column vectors. The equation Ax=b is consistent if and only if b is in the column space of A. This essentially means ...

For the updated question (is there a relation between k_x and k_y besides k_x=k_y which ensures that k_x\cdot x<k_y\cdot y\to x\neq y?), the answer is no. Suppose k_x\neq k_y. Then either k_x<k_y ...

Recall that for a multinomial distribution the PDF is f(x_1, \ldots, x_k; n, \pi_1, \ldots, \pi_k) = \frac{n!}{x_1!\cdots x_k!}\pi^{x_1} \cdots \pi^{x_k} = {{n} \choose {x_1,\ldots,x_k}}\pi^{x_1} \cdots \pi^{x_k} ...

You can write a=\cos\phi/r and b=\sin\phi/r. Then the expression you want to minimize reduces to \sum_{i} \left(x_{i} \cos\phi + y_{i} \sin\phi - r\right)^2\;. The equation you get from ...

For the first equation \text{(1) }\left(2x-y^{2}\right)\cdot y'=2y It's far easier to use x'=\frac {dx}{dy} instead \left(2x-y^{2}\right)\frac {dy}{dx}=2y \implies \left(2x-y^{2}\right)=2y\frac {dx}{dy} ...