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Questions tagged [inner-products]

For questions about inner products and inner product spaces, including questions about the dot product.

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Show that the inner product of two square summable series converges [closed]

Given that x and y are convergent square summable series, show that the inner product converges as well, i.e. the sum to infinity of xy
Name's user avatar
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a Cauchy Schwarz application [duplicate]

Here an inequality that I feel that CS may prove it but I can't find the right way to use it : $\left (\sum_{i=1}^{n}a_{i}\right )^{2}+\left (\sum_{i=1}^{n}b_{i}\right )^{2}\leqslant \left (\sum_{i=1}^...
Loca's user avatar
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True or False: Inner product on $\mathbb{R}^2$ satisfying a specific norm.

Verify or refute: There exists an inner product in $\mathbb{R}^2$ such that the norm of every vector $v=(v_1,v_2)$ is $\|v\|=|v_1|+|v_2|$. I think this is untrue. So I took $v=(1,0), y=(0,1)$. After ...
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Orthogonal projection is bounded

Definition: Let $U$ be a subspace of $V$. The orthogonal projection of $V$ onto $U$ is the operator $P_U\in L(V)$ given by $$P_U(u+w)=u$$ if $u\in U, w\in U^{\perp}$. Let $V$ be a space with inner ...
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Inner products induced by inner product and multiplication of vectors, e.g. $f(A,B) = tr(A^TB)-tr(A)tr(B)$ as an inner product?

It just came to my attention that both the expected value $\mathbb{E}(XY)$ and the covariance $\text{Cov}(X,Y)$ can be understood as scalar products. This is a consequence of the linearity of the ...
theta_phi's user avatar
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Are there self-adjoint operators with eigenvalues 1, -1 that are not isometries?

Let $f$ be a self-adjoint operator in an euclidean metric vector space $(V, g)$. a) Prove that the following are equivalent. i) The only eigenvalues of $f$ are 0, 1. ii) $f$ is an orthogonal ...
MrGran's user avatar
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Help with proof of Curl Double Product identity using Geometric Algebra. Most things seem to fall in place, but having a few issues.

So I'm pretty new to GA/Clifford Algebras, but it's been fairly interesting so far. I figured I'd try to prove some basic vector calculus identities with it, just to help me get my bearings. I decided ...
Copywright's user avatar
2 votes
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39 views

Inner Product Spaces Exercise

I have this exercise: "Let $M$ is a CLOSED subspace of Hilbert space $H$. Prove that $x\perp M \Leftrightarrow \left \| x \right \|_{H}=d(x,M)=\underset{y\in M}{\inf}\left \| x-y \right \|_{H}$. ...
Trần Nguyên Khang's user avatar
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Does linear operator of range $n \in \Bbb{N}$ conserve inner product?

Let $H_1, H_2$ be Hilbert spaces and $T \in \mathcal{L} (H_1, H_2)$ such that $\dim T(H_1) = n \in \Bbb{N}$ . There exists $\{x_k\}_{k=1}^n \subset H_1$ and $\{y_k\}_{k=1}^n \subset H_2$ such that for ...
Superdivinidad's user avatar
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Function-vector dualism of inner product

I have a question related to the dualism between the inner product of infinite-dimensional vectors and the integral over the product of their corresponding function representations, which is given by $...
Richard Schömig's user avatar
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How to determine all symmetric bilinear and sesquilinear forms

Let $( T )$ be the rotation with matrix representation $ T=\left[\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right] $ (a) Determine all symmetric bilinear forms $ \langle x, y\rangle $ on $\...
asdfgh jkl's user avatar
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Real Valued Polynomials and Inner Product

Let $P_n$ denote the space of real-valued polynomials of degree less than or equal to $n$. For which values of $n$ does $$ \langle f, g\rangle = f(1)g(1) + f(2)g(2) + f(3)g(3) $$ define an inner ...
nasiedlak's user avatar
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Linear Algebra: Orthogonal basis to find proj of $\mathbf{w}$ onto $\mathbf{y}$?

I'm currently studying for my final exam for linear algebra, and I'm a bit confused about how to find the projection of $\mathbf{w}$ onto $\mathbf{y}$. I already found the orthogonal basis for $W$, ...
ejry's user avatar
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Minimizing the resultant of two vectors where one vector is magnified by some $b$

Consider two vectors $\vec{r},\vec{s}$ in $ℝ^3$ with a constant $b$ chosen such that the length of the resultant $\vec{r}$ and $b\vec{s}$ is minimum. Prove that this occurs when $(\vec{r}+b\vec{s}),b\...
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1 answer
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Consequences of definition of scalar product

Definition: Let $V$ be a vector space over the field $K=\mathbb{R}$ (or over $K=\mathbb{C})$. The scalar product on $V$ is a function $V\times V\to K,$ denoted by $(x,y)\mapsto \langle x,y\rangle$, ...
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What is correct option regarding $ax+b,x+c$ orthogonal to each other...

Let $\mathbb{R}$ be a field of real numbers and V is the vectorspace of $P(\mathbb{R})$ of degree almost one. consider the bilinear form $$<>: V\times V\to \mathbb{R}$$ such that $$<f,g>=\...
math student's user avatar
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Existence of an inner product compatible with two decompositions of a vector space.

Let $V$ be a finite dimensional vector space. Then, given a decomposition $V = A \oplus B$, any inner product such that $A$ and $B$ are orthogonal is of the form $g = g_A \oplus g_B$ with a metric $...
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Next step in proof for inner product

I'm working on understanding the use of the properties of inner products better. And in particular I'm trying to understand this problem (problem 1.7 from Elementary Functional Analysis by B. Maccluer)...
loginfx's user avatar
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Bilinear forms, endomorphisms and skew symmetric matrices

Let $(V, g)$ be an inner product space and $f$ and endomorphism of $V$ such that $$ g(u,f(v)) = -g(f(u),v),\ \ \text{for all}\ u,v \in V. $$ Prove that a) $\ker(f)$ and $Im (f)$ are orthogonal ...
MrGran's user avatar
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Calculus of scalar products

In a certain problem I have been given the following data: $ds^2 = u^2du^2 + 2ududv + 2dv^2$, which would be the First Fundamental Form of an unknown surface, given a parametrization $\alpha(u,v)$. I ...
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Inner product spaces - Convergent [closed]

I have a question. I still worry about this definition. Let E is an inner product space $(E,\left \langle \cdot , \cdot \right \rangle)$. A sequence $(x_n)_{n\in \mathbb{N}=\left\{1,2,3,...\right\}}$ ...
Trần Nguyên Khang's user avatar
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An skew-symmetric matrix with respect to an orthonormal basis

Let $(V, g)$ be an euclidean vector space and $f$ and endomorphism of $V$ such that $$ g(f(u), v) = -g(u, f(v)),\ \text{for all}\ u, v \in V. $$ a) Prove that $\ker f$ and $im f$ are orthogonal ...
MrGran's user avatar
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can inner product of 2 different vectors in hilbert space be undefined? [duplicate]

ok so my question is simple If we take a Hilbert space, in my case I am taking an infinite dimensional complex numbered Hilbert space . ok, then how do we know that the inner product of any 2 ...
abx_pradB's user avatar
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$\inf_{a, b \in \mathbb{R}} \left( (1-a-b)^2 + (1-3a+b)^2 + (1-2a-b)^2 \right)$

Find $y = \inf_{a, b \in \mathbb{R}} \left( (1-a-b)^2 + (1-3a+b)^2 + (1-2a-b)^2 \right)$. This is a problem is in a course of biliear algebra and scalar products. Since it's about an inf, I thought ...
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Expectation of $u^\top(u + Ax)$, when $A$ and $u$ are nonlinear functions of $x$

Let $x\in\mathbb R^d$, and $s=\operatorname{softmax}(x)$. Let $y$ be a fixed one-hot vector such that $$u = s-y \\ v =(\operatorname{diag}(s) - ss^\top)x$$ I am interested in the inequality $u^\top (u ...
Phoenix's user avatar
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Matrix of orthogonal Projection onto a Hyperplane in $\mathbb{R}^n$

We consider the Euclidean space $\mathbb{R}^n$ equipped with the usual scalar product. Let $H = \{ x \in \mathbb{R}^n \, | \, a_1x_1 + \ldots + a_nx_n = 0 \}$, where $a_1, \ldots, a_n$ are given real ...
Denis's user avatar
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3 answers
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Why is $ R(A^*) \perp N(A)$ true?

Let a matrix the $A \in M_{n\times n}(\mathbb{C})$. My question is: (1) Why every matrix $A$ satisfies $ R(A^*) \perp N(A)$(where $R(A),N(A)$ are range of $A$,null space of $A$ respectively)? And why ...
David's user avatar
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0 answers
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Can angles be defined by norms that are not induced by inner products?

Can angles be (well-)defined in a normed vector space where the parallelogram law does not hold? In other words, if the norm is not induced by an inner product in a normed space (say $L^1$ space), can ...
chaohuang's user avatar
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1 answer
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Projection of vector

The projection of a vector $x$ onto a vector $u$ is $proj_u(x) =\frac{\langle x, u \rangle}{\langle u, u \rangle}u.$ Projection onto $u$ is given by matrix multiplication $proj_u(x)=Px$ where $P=\frac{...
David's user avatar
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1 vote
1 answer
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The inner product of column vectors is the same as matrix multiplication

I am very much new on the topic of inner product: Definition. The inner product of vectors $x, y \in \mathbb{R}^n$ is $\langle x, y\rangle =\sum_{i=1}^{n} x_ky_k=x_1y_1+x_2y_2+\dots+x_ny_n$ I can't ...
David's user avatar
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2 votes
1 answer
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Matrix of an Inner Product and Spectral Theorem

My linear algebra has become very rusty and now I've confused myself entirely. Let $V$ be an inner product space over an $n$-dimensional real vector space $V$. Moreover, let the set of vectors $$\...
Algebro1000's user avatar
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Tangent vectors to regular curve at constant angle to a fixed vector

Problem: Show that the tangent vectors to the regular curve $x(t)=(3t,3t^2,2t^3)$ make a constant angle ($\theta$) with the vector $a:=(1,0,1)$. We have $b:=x'(t)=(3,6t,6t^2)$. I worked out that $b\...
Alex Petzke's user avatar
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1 answer
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Orthogonal orthornomal bases imply pair-orthogonal vectors

While self-studying linear algebra i started thinking about following problem: Let's say that $A, B \in \mathbb{C}_{n\times n}$ are orthogonal in a Frobenius sense orthonormal bases of complex vector ...
Freechoice guy's user avatar
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1 answer
203 views

Proving $\|v\|^2 \ge \sum _{i=1}^n \langle v,e_i\rangle^2$

Prove that: $\|v\|^2 \ge \sum _{i=1}^n \langle v,e_i\rangle^2$ for any $v \in V$, where $V$ is an inner product space and $S = \{e_1, e_2, \ldots , e_n\}$ is an orthonormal subset of $V$. I know ...
CountDOOKU's user avatar
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0 votes
1 answer
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Space of square-integrable functions and its scalar product [closed]

For square integrable functions we define a scalar product: $$\int{d^3x \psi^* (\vec{x}) \phi (\vec{x})}$$ for any $\psi, \phi \in L^2(V) $. How can I show that the fundamental property $(\psi,\phi)=(\...
Dayane 's user avatar
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0 answers
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Inner product in RKHS

I am reading a paper and am confused by an expression about the inner product. It says that "Given a scalar-valued RKHS $\mathcal{H}$ with a positive definite kernel $k(x,x')$, $\cdots$ and $<\...
user1168149's user avatar
5 votes
0 answers
154 views

Does $X \times \mathbb{R} \simeq X$ hold for infinite dimension inner product space $X$?

The $X$ is infinite dimension real inner product space, not restricting it as a Hilbert space. This question has troubled my friend and me a lot of days. It is obviously true when $X$ is infinite ...
CTuser_103's user avatar
1 vote
1 answer
34 views

Is the Ball-Multiplier for the Fourier Transform auto-adjoint?

Let $f\in L^2(\mathbb{R})$, we define the ball multiplier as the operator $$Sf(x) := \int_{|\xi|<1} \hat{f}(\xi)e^{2\pi i x\cdot\xi}d\xi $$ where $$\hat{f}(\xi) := \int_{\mathbb{R}}f(y)e^{-2\pi i y\...
Mr_ion77's user avatar
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Prove $((x_n, y_n))$ converges to $(x, y)$ $\in X × Y$ if and only if $(x_n)$ converges to x in X and $(y_n)$ converges to y in Y.

$(X, ⟨·, ·⟩′)$ and $(Y, ⟨·, ·⟩′′)$ are inner product spaces over the same field, $F$. $X × Y$ is a vector field over $F$ with addition and scalar multiplication defined by $(x,\,y) + (x_1,\,y_1) = (x +...
number8's user avatar
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The positive/negative sign in inner product involve diagonal matrix and basis transformation matrix.

The expression of one inner product is $<\pi PDEP^{-1}, \pi PD^2EP^{-1}>$. Here, $\pi$ is a row vector that has strictly non-negative terms, D is a diagonal matrix with non-positive diagonal ...
Puning Wang's user avatar
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Effect of Basis Change on Absolute Vector Magnitudes

Does Basis Change Affect the Absolute Magnitude of Vectors? How does a change in basis impact the absolute length of vectors? I'm trying to understand the effect of a basis change on the absolute ...
fatFeather's user avatar
6 votes
2 answers
284 views

Is a scalar presented as a matrix or not here?

In the linear algebra course I am taking, the inner product of 2 vectors $\langle u, v \rangle$ is defined as being a scalar; however, it is also viewed as being a product of 2 matrices as $u^Tv$, as ...
Princess Mia's user avatar
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1 vote
2 answers
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How should we characterize the relationship between two matrix representations of a linear operator with respect to two different orthonormal bases?

Nielsen / Chuang remark on page 71 of "Quantum Computation and Quantum Information" that, if $| v_i \rangle$ and $|w_i \rangle$ are orthonormal bases, then the operator $U$ defined by $\sum_{...
mchk's user avatar
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Continuity on countably-normed Hilbert spaces

i was studying some Quantum Mechanics from this doctorate's work http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf and ata certain point, in Proposition 2 pag. 166 he means to prove the continuity of an ...
Marco Lugarà's user avatar
1 vote
1 answer
36 views

How can an identical vector have a lower dot product value than two different vectors?

Given two vectors: $\mathbf{v}_1 = (2, 2)$ and $\mathbf{v}_2 = (3, 3)$: Since the dot product is one method for measuring the similarity between vectors, and given that $\mathbf{v}_1 \cdot \mathbf{v}...
Edmar Miyake's user avatar
4 votes
2 answers
76 views

If $U $ is a unitary linear operator, how can I show that any matrix representation of $U$ must be a unitary matrix?

Nielsen / Chuang "Quantum Computation and Quantum Information" states on p. 70: "A matrix $ U$ is said to be unitary if $U^\dagger U = I$. Similarly, an operator $U$ is unitary if $U^\...
mchk's user avatar
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0 answers
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To find number of pairs of non-orthogonal vectors in a given set

I am quite stuck on this question. Any ideas/hints on how to approach will be helpful. Q. Let $E$ be an $n$-dimensional inner-product space over $\mathbb{R}$. Let $〈\cdot,\cdot〉$ denote the inner ...
ogirkar's user avatar
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1 vote
1 answer
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Can be use $u$-substitution for calculating the adjoint of an operator in Schwartz space?

I only have seen that for calculating the adjoint of an operator in $\mathcal S$, it used integration by parts, but I was thinking that if one can use substitution to find the adjoint. For exmple, for ...
Daniel Muñoz's user avatar
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0 answers
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Is this summation over Fourier coefficients a meaningful way to compute dot product despite its weaknesses? Or what is the better way to define it?

So let's say we have two vectors $a \in \mathbb R^n$ and $b \in \mathbb R^m$. Let $L=\text{lcm}(n,m)$. Consider the natural inclusions $a,b \mapsto \mathbb R^L$, which simply continually concatenate $...
Snared's user avatar
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4 votes
3 answers
268 views

Solving progressive tax calculation for pre-tax income

Progressive Tax Rate Explanation Progressive taxation works by taxing income within a certain bracket at different rates. For example: Bracket # % tax rate within bracket Min amount (exclusive) Max ...
Josh Sherick's user avatar

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