We can choose different bases to define different inner products. However, different orthonormal bases will generate the same inner product.
Could anyone explain the last sentence? Why different orthonormal bases give the same inner product?
2026-03-25 08:07:53.1774426073
Orthonormal Basis and Hermitian inner product
161 Views Asked by user539442 https://math.techqa.club/user/user539442/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 INNER-PRODUCTS
- Inner Product Same for all Inputs
- How does one define an inner product on the space $V=\mathbb{Q}_p^n$?
- Inner Product Uniqueness
- Is the natural norm on the exterior algebra submultiplicative?
- Norm_1 and dot product
- Is Hilbert space a Normed Space or a Inner Product Space? Or it have to be both at the same time?
- Orthonormal set and linear independence
- Inner product space and orthogonal complement
- Which Matrix is an Inner Product
- Proof Verification: $\left\|v-\frac{v}{\|v\|}\right\|= \min\{\|v-u\|:u\in S\}$
Related Questions in ORTHONORMAL
- Orthonormal basis for $L^2(\mathbb{R}^n,\mathbb{F})$
- What is $\| f \|$ where $f(x)=\sum\limits_{n=1}^\infty \frac{1}{3^n} \langle x,e_n\rangle$
- Forming an orthonormal basis with these independent vectors
- Orthogonal Function Dirac Delta Series
- Sum of two rank $1$ matrices with some property gives rank $2$ matrix
- Zero element in an Hilbert space is orthogonal?
- Prove that $\lVert X\rVert^2 =\sum_{i,j=1}^\infty\lvert\langle u_i,Xu_j\rangle\rvert^2$.
- Is there any connection between the fact that a set of vectors are mutually orthogonal and the same set of vectors are linearly independent
- Compute the norm of a linear operator using a normal basis in an infinite Hilbert space
- If $M$ is the span of a finite orthonormal set in a Hilbert space then $M$ is closed
Related Questions in CHANGE-OF-BASIS
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- What is meant by input and output bases?
- Change of Basis of Matrix: Two points of view
- Change of Basis (Transformation Matrix)
- Diagonalization and change of basis
- Change of Basis Matrix. Doubt about notation.
- Why does the method of getting a transition matrix seems reversed to me?
- Finding bases to GF($2^m$) over GF($2$)
- Block diagonalizing a Hermitian matrix
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?
Actually, any basis defines the usual inner product in terms of coordinates, so that it actually becomes an orthonormal basis.
Now, if an inner product is given, and $b_1,\dots, b_n$ is an orthonormal basis, its inner product will coincide with the original one, because $$\left\langle\sum_i\alpha_ib_i,\, \sum_i\beta_ib_i\right\rangle\ =\ \sum_i\alpha_i\beta_i$$