I found statement of theorem, that for any field $F$, any homomorphism $f:GL(n, F)\rightarrow F^{*}$ is composition $f=g\circ det$ for some $g:F^{*}\rightarrow F^{*}$ - endomorphism, and $det$ - determinant. But i could'n find proof of that theorem, could you help?
2026-03-25 09:50:17.1774432217
Any homomorphism from $GL(n, F)$ to $F$ is composition of $det$ and $F$-endomorphism.
47 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in LINEAR-ALGEBRA
- An underdetermined system derived for rotated coordinate system
- How to prove the following equality with matrix norm?
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- Why the derivative of $T(\gamma(s))$ is $T$ if this composition is not a linear transformation?
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- Summation in subsets
- $C=AB-BA$. If $CA=AC$, then $C$ is not invertible.
- Basis of span in $R^4$
- Prove if A is regular skew symmetric, I+A is regular (with obstacles)
Related Questions in ABSTRACT-ALGEBRA
- Feel lost in the scheme of the reducibility of polynomials over $\Bbb Z$ or $\Bbb Q$
- Integral Domain and Degree of Polynomials in $R[X]$
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- Group with order $pq$ has subgroups of order $p$ and $q$
- A commutative ring is prime if and only if it is a domain.
- Conjugacy class formula
- Find gcd and invertible elements of a ring.
- Extending a linear action to monomials of higher degree
- polynomial remainder theorem proof, is it legit?
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
Related Questions in DETERMINANT
- Form square matrix out of a non square matrix to calculate determinant
- Let $T:V\to W$ on finite dimensional vector spaces, is it possible to use the determinant to determine that $T$ is invertible.
- Optimization over images of column-orthogonal matrices through rotations and reflections
- Effect of adding a zero row and column on the eigenvalues of a matrix
- Geometric intuition behind determinant properties
- Help with proof or counterexample: $A^3=0 \implies I_n+A$ is invertible
- Prove that every matrix $\in\mathbb{R}^{3\times3}$ with determinant equal 6 can be written as $AB$, when $|B|=1$ and $A$ is the given matrix.
- Properties of determinant exponent
- How to determine the characteristic polynomial of the $4\times4$ real matrix of ones?
- The determinant of the sum of a positive definite matrix with a symmetric singular matrix
Related Questions in GENERAL-LINEAR-GROUP
- Proving that $Aw\in \langle w\rangle \implies A$ is of the form $\lambda I_n$
- Can we find element of order $q^2-1$ in $\text{GL}_2(\mathbb{F}_q)$?
- Is $H$ is a normal subgroup of $G$? Yes/NO
- Action of $GL(\Bbb F_2^3)$ on the sub-spaces of $\Bbb F_2^3$ of dimension $2$
- The orbit space GL(n,R)/O(n)
- Can any finite group of order $n$ be embedded in $SL_n(\mathbb{Z})$?
- eigenvectors of invertible matrices over the complex numbers
- eigenspace of the power of a matrix
- Isomorphism between $GL_2(\mathbb{F}_2)$ and $S_3$
- Center for a subgroup of $\operatorname{GL}_{2}(\mathbb{F_{7}})$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?