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Questions tagged [inner-product-space]

An inner product space is a vector space equipped with an inner product. The inner product is a generalization of the “dot” product often used in vector calculus.

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Confusion on the Gram-Schmidt process for complex vectors

I am having some trouble with the inner product and the Gram-Schmidt process for complex vectors as I am trying to learn it on my own. This is mainly due to the discrepancy with my text book and what ...
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How to find the vector $(\gamma_1,\gamma_2,\delta)^T$ that characterizes the matrix and product?

I have the matrix: $$H=\left[\begin{matrix}\gamma_1&\delta\\\delta&\gamma_2\end{matrix}\right]$$ and the inner-product: $$<x,y>=x^THy,$$ and has to find the vector $(\gamma_1, \gamma_2, \...
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The question is incorrect?

Given an Inner product space $V$ over a field $F = \mathbb{R}$ or $F = \mathbb{C}$ (the question should work for both) and $\phi : V \to F$ is a functional, and also $< \cdot, \cdot>$, and an ...
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Corollary of Projection onto a closed convex set and geometric interpretation

I need help with geometric interpretation of this theorem and with the corollary of the theorem: Theorem: projection onto a closed convex set Let $K \subset H$ be a nonempty closet convex set. ...
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Fourier transform on $\mathbb{Z}_{2}^{d}$

Let $\mathbb{Z}_{2}^{d} = {\{\textbf{t} = (t_1, \ldots, t_d) : t_j \in \mathbb{Z}_2}\}$. Define the inner product on functions $f, g : \mathbb{Z}_{2}^{d} \rightarrow \mathbb{C}$ to be: $$\langle f, ...
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Linear Algebra Scalar and Vector Projection [on hold]

I have a problem with one part of a question. I just need help with part b. How would I go about applying the equation to find the scalar and vector projection of v2 onto v1? Thanks!
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Why is linearity not satisfied for this inner product?

The inner product given is $\langle u,v \rangle = u_{1}^{2}v_{1}^{2} + u_{2}^{2}v_{2}^{2}$. If I am thinking of linearity correctly, then all that means is the coefficients in front of each like term ...
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Can the eigenvectors of a linear operator in an infinite-dimensional space span the space and be linearly dependent at the same time?

Consider a vector space $V$ over the complex field which is infinite-dimensional with a Euclidean inner-product. Let $L$ be a linear operator on $V$. Say a subset of eigenvectors of $L$ forms a ...
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SVD with non-standard inner product

I have a linear transformation $T: V \to W$ where $V$ and $W$ are finitely generated real inner product spaces and their inner product is not necessarily standard. I also have $K: V \to V$. My goal is ...
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Finding algebraic expression of a parallepiped given directions and side lengths

Let $u_1,\dots,u_n$ be linearly independent unit vectors in $\mathbb R^n$. Let $T$ be the parallepiped centred at $0$, with sides parallel to $u_i$ and side lengths $l_i$, $i=1,2,\dots,n$. My question ...
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Inner product on diffential forms independence ortonormal basis

Suppose $\{e_1,...,e_n\}$ is a positive orthonormal basis for the tangent space at a point $p$ in an oriented n-manifold $M$, then define the inner product on $\Omega^k(M)$, for each $k$, by: $$\...
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Why is the first one not an inner product but the second one is?

$\langle p,q \rangle = \int_{-1}^1 p^\ast(t)q(t)tdt$ over $\mathbb{P}$ $\langle p,q \rangle = \int_{-1}^1 p^\ast(t)q(t)(t+1)dt$ over $\mathbb{P}$ I believe positivity works for both of them, and I ...
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Is the expectation of the inner product of two random vectors the inner product of the expectation of each one individually?

Am trying to figure this out. Think the answer might depend on the specific spaces on operators involved, but any help you could give would be much appreciated!
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What is the meaning of $\frac{(Ay,x)}{(y,x)}$?

$x^TAy$ is the inner product of a matrix A. If x,y are unit vectors, then what is the meaning of $\frac{x^TAy}{x^Ty}$? What does it do to the inner product? The orthogonal component of y wrt x is ...
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(real symmetric matrices) does orthogonality of eigenvectors (distinct eigenvalues) depend on choice of basis?

For some reason I cannot wrap my head around this one. My schoolbook begins the chapter on eigendecomposition of real symmetric matrices by stating that eigenvectors from distinct eigenspaces are ...
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Orthogonal diagonalization without eigenvectors

I stumbled onto a method for orthogonally diagonalizing a symmetric matrix with real entries and I was wondering what advantages (if any at all) it has over the eigenvector method. It hinges on the ...
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$\langle x,y \rangle = 0 \iff \forall \alpha\in\Bbb R, |x|\le|x+\alpha y|$

If $V$ is a inner product space and $x,y\in V$, why is $\langle x,y \rangle = 0$ equivalent to say that for every $\alpha\in\Bbb R$, $|x|\le|x+\alpha y|$? I understand that $\langle x,y \rangle = 0$ ...
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Inner products of of two vectors which are both orthogonal to a third vector.

If, in an $R^n$ space, $(x,y)=0$ $(x,z)=0$ Then what about $(z,y)$? What if $(z,y)\approx 1$ (although z,y are two different vectors with different elements), then what can we say about y and z? ...
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Orthogonal Projection and Inner Product Space

Prove: Let V be an inner product space. $W⊂V$ and $v∈V$. Let $w∈W$ be an orthogonal projection. Then for every $u∈W ; ||w-v||<= ||u-v||$. I really do not have a clue on how to solve this.
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Inner product that makes vectors an orthonormal basis

Let $X= \begin{pmatrix} a \\ b \end{pmatrix} $ and $Y=\begin{pmatrix} c \\ d \end{pmatrix}$ be two vectors in the plane. Do we have the existence of an inner product that makes ...
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Does the inner product of a matrix, $\frac{x^TAy}{x^Ty}$ stay the same or fall in a range for any x,y?

Is there any bound on $\frac{x^TAy}{x^Ty}$, for any vector $x$? I am observing that $\frac{x^TAy}{x^Ty}$ is approximately the same even when I change $x$. Why is that? Is there any property for the ...
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Inner Product & Linear Map property (from Linear Algebra Done Right, 7C question 4)

Might be a silly question, but I am unsure about how this solution got the equality $\langle T^*Tv, v \rangle = \langle Tv, (T^*)^*v \rangle$ as in, how we can move the linear map, $T$, around like ...
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Write $(-1,1,2,2)^T$ in terms of the basis $U$

I have found that $U=\{1/2\pmatrix{1\\1\\1\\1},1/2\pmatrix{-1\\-1\\1\\1},\frac{1}{\sqrt{2}}\pmatrix{-1\\1\\0\\0},\frac{1}{\sqrt{2}}\pmatrix{0\\0\\1\\-1}\}$ is an orthogonal basis for $\mathbb{R}^4$. ...
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Are all (real) inner products symmetric bilinear forms?

Given that an inner product on a real vector space $V$ is a function $b : V \times V \rightarrow \mathbb{R}$ satisfying: $b$ is bilinear (that is, $b$ is linear in the first variable when the second ...
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1answer
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Find the matrix of inner product space with orthonormal basis

Let $\mathbb{R}^n,\; n\geq 2$ be equipped with standard inner product. Let $\{v_1,v_2,......,v_n\}$ be $n$ column vectors forming an orthonormal basis of $\mathbb{R}^n$. Let $A$ be the $n\times n$ ...
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43 views

Proof: If $\mathbb{R}^{n}=S_{1}\bigoplus S_{2}$ then $S_{1}\bigcap S_{2}=\left \{ \mathbf{0}\right \}$

Here's a question that I've been working on from my Linear Algebra textbook (Larson, 8th ed.) and I'd like to check if what I've been doing is correct. Question: Prove that if $S_{1}$ and $S_{2}$ ...
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If {$u_{i}$} is an orthonormal basis, then $proj_{s}v = ( v\cdot u_{1} )u_{1} +\cdots+ ( v\cdot u_{t} )u_{t}$

I'm working on a proving problem in my Linear Algebra textbook (Larson, 8th ed.) and I'd like to ask for help in finishing the proof for a theorem on the projection onto a subspace. Prove: If $\left \...
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Is the integral of the product of two real-valued functions an inner product for every kind of integral?

I'm wondering if the integral of the product of two real-valued functions on a given space is also necessarily also an inner product for every function that can be defined on that space. E.g., are the ...
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$l_p : (p≠2)$ is not an inner product.

I am using Kreyszig's functional analysis it is example in it which says that $l_p : (p≠2)$ is not an inner product space. For proof it uses the standard norm on $l_p$ and shows that it does not ...
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Prove that the set $W^{\perp}$ is a subspace of $V$ and use this to find $W^{\perp}$ when $W$ is the span of $(1,2,3)$ in $V=\mathbb{R}^{3}$

My Linear Algebra book (Larson, Eight Edition) has a two-part exercise that I'm trying to answer. I was able to do the first [proving] part on my own but need help tackling the second part of the ...
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Is there an elegant way to define orthogonality (and/or angles) without inner products, metrics, or norms?

I was wondering if there is an elegant and intrinsic way to define orthogonality on vectors without introducing inner products? Obviously "elegance" is subjective, so I'll try and give a sketch of the ...
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Eigenvalues of a linear combination of projections

So I have an euclidean or unitary vector space with two linearly independent, but not necessarily perpendicular vectors $x$ and $y$. $P_x$ and $P_y$ are the orthogonal projection operators to $L(x)$ ...
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inner product vector space Book?

I am looking for a book or lecture course on inner product (euclidean) vector space with problems; please help!
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Orthonormal basis of eigenvectors of a self-adjoint operator

Let $V$ be a finite-dimensional real inner product space and suppose $T$ is an endomorphism on $V$. Show that $(T+T^*)/2$ is self-adjoint and show that there is an orthonormal basis $\{\vec{v_1},...,\...
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$M=(M^\bot)^\bot$ if and only if $M$ is a subspace [duplicate]

Show that a subset $M$ of a finite-dimensional inner product space satisfies $M=(M^\bot)^\bot$ if and only if $M$ is a subspace. I am having trouble proving $M$ is a subspace $\implies$ $M=(M^\bot)^\...
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1answer
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Intersection and sum of closed sets and their orthogonal complements

I don't understand in the answer of 11b, why does that result follows from 11a? I know that the orthogonal complement is always a closed set, and that for closed sets, the orthogonal complement of ...
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2answers
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Prove that $\|u + v\| =\|u\| + \|v\|$ if and only if $u$ and $v$ have the same direction.

I'd like some help with a proving problem I have in my linear algebra textbook. My background: I'm a business student taking a linear algebra class and I have a really hard time with proving problems ...
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1answer
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For what values of c is ||c(1,2,3)|| = 1?

I just wanted to check if what I'm doing for a particular problem is correct. My background: business student with average math skills taking a linear algebra class. Would appreciate it if someone ...
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1answer
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Confusion about a form $<,>$ on a finite-dimensional vector space $V$ being non-degenerate on a subspace $W$

I'm currently going through Artin's Abstract Algebra book, and I'm a bit confused about the following: He says that a form $<,>$ on a finite dimensional vector space $V$ is non-degenerate if ...
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Ordering of inner products with constraints and matrix multiplication

Suppose that $x^T y \ge z^T y$ for all vectors $x \in \mathbb{R}^d$. If $H \in \mathbb{R}^{d \times d}$ is symmetric positive definite, then $x^T H y \ge z^T H y$. Approach. If $y_i \ge 0$ then we ...
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Inverse Problem for positive definite matrices

Does there exists a positive definite matrix $A^H = A\in \mathbb C^{n \times n}$ under some condition such that for a given $x,y \in \mathbb C^n$, where $x \neq 0$. $Ax = y$ holds good. What I have ...
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Why is every scalar product equivalent to the Standard Scalar Product on $\mathbb{R}^2$

I recently encountered the following claim: Consider $\mathbb{R}^2$ endowed with an arbitrary scalar product. By choosing an orthonormal basis we may assume that the given scalar product is the ...
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For vectors in an Inner product space, would this be a valid proof for the triangle inequality?

Let $V$ be an inner product space. Show that $$||x + y|| ≤ ||x|| + ||y||$$ for all x, y ∈ V By Pythagoras’ Theorem $$||x + y||^2 = ||x||^2 + ||y||^2$$ $$||x + y||^2 = (||x|| + ||y||)^2-2||x||||y||$$ ...
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Is Orthogonal Projection independent of basis for any Basis?

Assuming the usual inner product $\langle x, y\rangle = \bar x^\mathsf{T} y$ on a complex vector space $V$, and defining the projection $\mathbf P_W\colon V\to W\leqslant V$ as $$\mathbf P_W(v) = \...
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1answer
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Prove the eigenvectors of a reflection transformation are orthogonal

This is part of a problem to prove that all reflection transformations are diagonalizable. For $T : \mathcal{V} \to \mathcal{V} \iff T^2 = I$, I've shown (1) that cases where $T$ has only one ...
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Positive definite operators commutes with any linear operator

Suppose $V=\mathbb R^2$ (Just a special case), let $M=\begin{bmatrix}1 & 0 \\0 & 2 \end{bmatrix}$, $K=\begin{bmatrix}0 & 1 \\0 & 0 \end{bmatrix}$. Note that $M, K$ does not commute, $...
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1answer
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Proving Cauchy-Schwartz inequality without the vanish assumption of inner products

I've came accross the following question in a book that I'm studying from, about Hilbert spaces - And the answer is - What I don't understand is why implies that - ?
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Define inner product on dual space.

V is a Hilbert space By Riesz Representation Theorem: $\forall f\in V^*\exists v$ s.t $f=l_v $ where $l_v(x)=<x,v>$ and $||l_v||=||v||$(Using this fact can check that norm of dual space ...
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68 views

if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function $\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$ is convex in $R^n$

Show by direct estimates that if $ A \in R^{n \times n}$ , $A > 0$ and $ b \in R^n$ then the function $$\frac{1}{2}\langle Ax,x\rangle - \langle b,x\rangle$$ with $x$ is convex on $R^n$. My ...
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Is the norm of a vector given by dividing it by the square root of the inner product with itself?

I've known the norm (aka. unit vector in the direction of) a vector (let's call it $v$) to be given by the formula: $v/\sqrt{v \cdot v}$, where $\cdot$ is the dot product. But can this be ...