Proving that a language defined by a regular expression is equivalent to a right linear grammar

After thinking for a bit, I am not able to prove a double inclusion proof for the following problem. It seems very interesting to me.

Consider the regular expression $r= ((1(00)^∗1 + 0)1)^∗$ and the right-linear grammar $G= ({S,A},{0,1},S,P)$, where $P$ consists of the following rules:

$Srightarrow 1A|01S|lambda$

$Arightarrow 00A|11S$

Prove that $L(G)subseteq L(r)$ and vice versa.

linear algebra – Showing that a vector $xin K_{lambda}$ in generalized eigenspace contains zero values

So I’m currently studying linear algebra and I came across this question that asks to show that if
$x= (x_1,x_2,x_3,x_4,x_5) in K_{lambda_1}$ where $K_{lambda_1}=ker((T-lambda_{1}I)^3)$ such that $K_{lambda_1}$=Span${e_1,e_2,e_3}$, then $x_4=x_5=0$.

The proof that I currently have is that since $xin$ Span${e_1,e_2,e_3}$, $x$ can only be represented as a linear combination of $e_1,e_2,e_3$ through which we can see that $x_4$ and $x_5$ are always $0$ $therefore$ $x_4=x_5=0$ $forall xin K_{lambda_1}$

However, I’m not sure if the proof is valid and sufficiently shows that $x_4=x_5=0$.

Any tips or suggestions would be greatly thankful

linear algebra – Does Number of Pivots (Independent Vectors) Means Dimension of That Vector?

I’m confusing about one topic. Concider we have that matrix;

|1 1 2|

|2 1 3|

|3 1 4|

|4 1 5|

So we know that this vector has only 2 independent vector since col3=col1+col2 and it has 2 pivot. C(A) fills 2-D space inside 4-D space.

But how the vectors satisfying y=5 the equation form a 2-D space in 3-D?

Isn’t it this has only 1 pivot since (0 5 0) (x y z)? So why it’s not 1-D space in 3-D?

Could we prove the system of linear equation is consistent $iff$ the rank $=$ # of rows deducitvely?

Proof subject: a system of linear equation(SLE) $mathbf{Ax=b}$ is consistent $iff$ the rank of $mathbf{A}$ equals to the number of rows of $mathbf{A}$

Description:

  1. I want to prove this deductively but not inductively.
  2. I have read the proof of a SLE is consistent $iff$ rank$((textbf {A}))=$ rank$((textbf{A}|~textbf{b}))$. It uses a concept of dimension. Like, It’s in the dimension of $textbf A$, so the system must be consistent because $textbf b$ is image of $T_textbf A$. But I want other proofs without this dimension thing. or I want someone could give a proof unfolding this relation between dimension and consistency because for what I’ve concerned, it’s not straightforward enough as a proof by just saying if it’s in the dimension of $textbf A$, then blablabla….
  3. I have read a proof like: all of situation where the SLE is inconsistent must have a last zero row after eliminated into reduced Echelon Form. It’s a more inductive way and definitely not what I want.

linear algebra – $A$-invariant subspaces $Gr(k)$

Let $A in GL(d,mathbb{R})$, which is the space of $dtimes d$ invertible matrices. I want to find $A$-invariant subspaces $Gr(k)$, which is a Grassmanian of $k$-planes of $mathbb{R}^{d}$.

My attempt: I think one needs to find the eigenspaces $wedge^{k} A$ that are $A$-invariant subspaces $Gr(k)$. If the previous sentence is correct, I have the following problems:$A$-invariant subspaces $Gr(k)$ are $k-$dimentional, but the eigenspaces $wedge^{k} A$ are $binom nk$ dimensional.

Polynomial bijections $mathbb{Z}^2tomathbb{Z}^2$ are linear

Let $P, Qin mathbb{Z}(x, y)$ be polynomials such that the map $$mathbb{Z}^2tomathbb{Z}^2, : (x, y)to (P(x, y), Q(x, y))$$ is bijective.

Are $P$ and $Q$ linear?

linear algebra – Coordinate transformation can this be done with AffineTransform?

Looking for alternatives or improvements for accomplishing the following:

Given the position of points A, B, and C in both the x-y and u-v planes, determine transformation functions to map values between the x-y and u-v planes.
Points A and B are on the v axis.
Point C is on the u axis.enter image description here

Below is what I have developed so far.

  1. Recommendations for improvements are welcome.
  2. Can this the result is expressed as an AffineTransform? This implementation uses (r + v).m but the AffineTransform uses r.m + v, perhaps there is a way to revise the implementation?
(*example data*)
{Axy, Bxy, Cxy, 
   Pxy} = {{0.2, 0.8}, {0.1, 0.15}, {0.8, 0.25}, {0.6, 0.7}};
{Auv, Cuv} = {{0, 75},  {100, 0}};

(*the unit vectors for the u-v plane, w.r.t. the x-y coordinate*)
Vxy = Normalize( Axy - Bxy);
Uxy = Normalize({Last(Vxy), - 
    First(Vxy) }); (*axis orthogonal to Vxy*)
(*the origin of the u-v plane in x-y coordinates is the intersection 
of the v and u axis*)
UVoxy = First@(  
   Axy + k1 Vxy /. Solve( Axy + k1 Vxy == Cxy + k2 Uxy , {k1, k2} ) );

(*scale factors for u-v axis*)
Uscale =   First@Cuv / EuclideanDistance(UVoxy, Cxy);
Vscale =   Last@Auv / EuclideanDistance(UVoxy, Axy);
mUVscale = {{Uscale, 0}, {0, Vscale}};
(*rotation from x-y to u-v*)
r1 = RotationMatrix({{1, 0}, Uxy}) ;
(*combined rotation and scaling*)
mXYtoUV = r1 . mUVscale;
mUVtoXY = DiagonalMatrix((1 / Diagonal@mUVscale)) . Transpose(r1)  ;

(*test: original xy points to uv*)
testUV =  (# - UVoxy) . mXYtoUV & /@ {Axy, Bxy, Cxy, Pxy} ;

(*test: computed uv points to xy*)
testXY01 = (# . mUVtoXY + UVoxy) & /@ testUV;

(*test: original uv points to xy*)
testXY02 = (# . mUVtoXY + UVoxy) & /@ {Auv,  Cuv} ;

(*report: xy to uv*)
Framed@Labeled(
  Grid({ { "pt", Style("UV data", Bold),  Style("UV computed", Bold)}
    , { Column({"A", "B", "C", "P" })
     , {Auv, {"-", "-"}, Cuv, {"-", "-"}} // MatrixForm 
      , testUV // MatrixForm
     } 
    }
   , Frame -> All), Style("xy to uv", 16, Bold), Top)

(*report: uv to xy*)
Framed@Labeled(
  Grid({ { "pt", Style("xy data", Bold),  Style("xy computed", Bold)}
    , { Column({"A", "B", "C", "P" })
     , {Axy, Bxy, Cxy, Pxy} // MatrixForm 
      , testXY01 // MatrixForm
     } 
    }
   , Frame -> All), Style("uv to xy", 16, Bold), Top)

results

results

Finally, here is the code to display a sketch of the system

(*sketch of the system*)
Graphics({AbsolutePointSize(10)
  , Point({Axy, Bxy, Cxy, Pxy})
  , AbsoluteThickness(1)
  , AbsoluteDashing(10)
  , Darker@Blue
  , Arrowheads({-0.05, +0.05})
  , Arrow({UVoxy - 0.35 Vxy, UVoxy + 0.65 Vxy})
  , Arrowheads(+0.05)
  , Arrow({UVoxy, UVoxy + 0.95 Uxy})
  , Darker@Green
  , Text(Style("A", Bold, 18), Axy + {0.03, 0.03}, {-1, 0})
  , Text(Style("B", Bold, 18), Bxy + {0.03, 0.03}, {-1, 0})
  , Text(Style("C", Bold, 18), Cxy + {0.03, 0.03}, {-1, 0})
  , Darker@Red
  , Text(Style("P", Bold, 18), Pxy + {0.03, 0.03}, {-1, 0})
  , Darker@Blue
  , Text(Style("u", Bold, Italic, 14), UVoxy + 0.95 Uxy , {-1, 0})
  , Text(Style("v", Bold, Italic, 14), UVoxy + 0.65 Vxy , {-1, -1})
  , Text(Style("-v", Bold, Italic, 14), UVoxy - 0.35 Vxy , {0.5, 1})
  }
 , Axes -> True
 , PlotRange -> {{-0.1, 1.2}, {-0.1, 1.1} }
 , Frame -> True
 )

linear algebra – Annihilator of direct product

Let $E_1,E_2$ be two subspaces in $GF(2)^n$ such that $E_1cap E_2={0}$. If $E_1times E_2={(x,y):xin E_1,yin E_2}$, is it true that $$(E_1times E_2)^{perp}=E_1^{perp}times E_2^{perp},$$
where $L^{perp}={xin GF(2)^n:langle x,yrangle =0, forall yin L}$ is the annihilator of a linear subspace $L$.

smoothness – Linear combinations of translated radial test functions

Let $n in mathbb{N}$, $Omega subseteq mathbb{R}^n$. Function $f colon Omega to mathbb{R}^n$ shall be called translated radial function if there exists $x in mathbb{R}^n$ and $g colon (0, infty) to mathbb{R}$ such that $forall y in Omega$ we have $f(y) = g( | x – y |)$, where $| cdot |$ denotes the Euclidean norm. Let $P( Omega)$ denote the family of all linear combinations of translated radial functions on $Omega$ which are test functions, that is, they are $C^infty$ smooth and of compact support.

I wasn’t able to find any papers which studied the properties of $P(Omega)$, though that is probably due to the fact that I’m not familiar with terminology that might’ve already been coined.

My questions are:

  1. If this family has already been studied (probably it was, I think), where should I look for papers regarding it?
  2. I’m especially interested in whether $P(Omega)$ is dense in $C^infty_c(Omega)$, the family of smooth functions with compact support with topology of uniform convergence of all derivatives on compact sets.
  3. If the answer to previous questions is true, I’m also interested in whether one can use a different norm (or even metric) to define what radial function means here and still have mentioned density.

linear algebra – How can one create a continuous function that transitions a 2d input through a

This is my first question here, hopefully it isn’t too bad. Apologies in advance if it isn’t formulated in a manner coherent with the expected formalism. Let me know if I need to make corrections and I will do so immediately.

I am trying to create a 3D function for which the first 2 (x;y) are parameters are contiguous of a 2D space. i.e. example 2d function graphs

and the third (z) acts to transition/”morph” the function that is applied to the prior 2 rational numbers through the possible relations between x and y.
A discrete (naive) implementation of this might be merely to instantiate a case statement

X(x;y;z) = { 
          |y|, for z=1 // absolute
          y^2, for z=2 // quadratic
          y^3, for z=3 // cubic
          ...
          x*y  for z=n
          }

Are there any methods by which one might transition the expression of the relation between x and y values with respect to a space parameterized by z. e.g

(identity(x;y) -> absolute(x;y) -> quadratic(x;y) -> cubic(x;y) .... // positive function (0<z)
-identity(x;y) <- -absolute(x;y) <- -quadratic(x;y) <- -cubic(x;y)) // negative functions (0>z)

How might one implement something like this?
Best regards and thanks in advance.