Is there any visual or not, interpretation of the determinant of a quadratic form? I think it helps me to understand why the determinant is not an invariant under a change of basis. Because the determinant of a matrix of a linear map means scaling the area of a parallelogram, so it doesn't matter which basis we chose.
2026-04-05 20:41:49.1775421709
Determinant of a quadratic form.
255 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related 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 LINEAR-TRANSFORMATIONS
- Unbounded linear operator, projection from graph not open
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- A different way to define homomorphism.
- Linear algebra: what is the purpose of passive transformation matrix?
- Find matrix representation based on two vector transformations
- Is $A$ satisfying ${A^2} = - I$ similar to $\left[ {\begin{smallmatrix} 0&I \\ { - I}&0 \end{smallmatrix}} \right]$?
- Let $T:V\to W$ on finite dimensional vector spaces, is it possible to use the determinant to determine that $T$ is invertible.
- Basis-free proof of the fact that traceless linear maps are sums of commutators
- Assuming that A is the matrix of a linear operator F in S find the matrix B of F in R
- For what $k$ is $g_k\circ f_k$ invertible?
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 QUADRATIC-FORMS
- Can we find $n$ Pythagorean triples with a common leg for any $n$?
- Questions on positivity of quadratic form with orthogonal constraints
- How does positive (semi)definiteness help with showing convexity of quadratic forms?
- Equivalence of integral primitive indefinite binary quadratic forms
- Signs of eigenvalues of $3$ by $3$ matrix
- Homogeneous quadratic in $n$ variables has nonzero singular point iff associated symmetric matrix has zero determinant.
- Trace form and totally real number fields
- Let $f(x) = x^\top Q \, x$, where $Q \in \mathbb R^{n×n}$ is NOT symmetric. Show that the Hessian is $H_f (x) = Q + Q^\top$
- Graph of curve defined by $3x^2+3y^2-2xy-2=0$
- Question on quadratic forms of dimension 3
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?
The fact that only the square class of the determinant is well-defined should tell you that it is not really a geometric, but rather an algebraic invariant. For instance if you work over the real numbers only its sign is well-defined.