Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
1answer
31 views

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 ...
0
votes
0answers
15 views

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 ...
0
votes
1answer
13 views

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 ...
1
vote
1answer
21 views

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: $$\...
0
votes
1answer
38 views

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 ...
0
votes
0answers
19 views

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!
2
votes
0answers
44 views

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 ...
0
votes
1answer
32 views

(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 ...
1
vote
1answer
40 views

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 ...
0
votes
2answers
40 views

$\langle x,y \rangle = 0 \iff \forall \alpha\in\Bbb R, |x|\le|x+\alpha y|$

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 $\langle x,y \rangle = 0$ ...
0
votes
1answer
17 views

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? ...
0
votes
0answers
18 views

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.
0
votes
2answers
18 views

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 ...
-4
votes
0answers
14 views

Which of the following vectors are orthogonal to each other with respect to the inner product [closed]

Which of the following vectors are orthogonal to each other with respect to the inner product The answer I was given is 4. When I worked out which vectors are orthogonal to each other I got A and B, ...
0
votes
1answer
37 views

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 ...
0
votes
1answer
21 views

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 ...
1
vote
2answers
36 views

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 $\mathbb{R}^4$. ...
-2
votes
1answer
26 views

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 ...
0
votes
1answer
33 views

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 $n\times n$ ...
1
vote
1answer
42 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 $S_{2}$ ...
0
votes
1answer
35 views

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 \...
0
votes
1answer
28 views

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 ...
0
votes
1answer
31 views

$l_p : (p≠2)$ 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 ...
0
votes
2answers
34 views

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 ...
0
votes
2answers
36 views

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 ...
-1
votes
0answers
30 views

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 $L(x)$ ...
1
vote
2answers
41 views

inner product vector space Book?

I am looking for a book or lecture course on inner product (euclidean) vector space with problems; please help!
0
votes
0answers
35 views

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},...,\...
0
votes
0answers
24 views

$M=(M^\bot)^\bot$ 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)^\...
0
votes
1answer
29 views

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 ...
2
votes
2answers
71 views

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 ...
0
votes
1answer
15 views

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 ...
0
votes
1answer
27 views

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 ...
1
vote
2answers
36 views

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 ...
0
votes
1answer
42 views

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 ...
0
votes
1answer
30 views

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 ...
0
votes
3answers
27 views

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 $$||x + y||^2 = ||x||^2 + ||y||^2$$ $$||x + y||^2 = (||x|| + ||y||)^2-2||x||||y||$$ ...
0
votes
2answers
56 views

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) = \...
2
votes
1answer
50 views

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 ...
0
votes
0answers
28 views

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, $...
0
votes
1answer
25 views

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 - ?
0
votes
1answer
19 views

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 ...
1
vote
1answer
66 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 ...
0
votes
0answers
26 views

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 ...
0
votes
0answers
10 views

Representing a transformation's matrix as an inner product

Let $V$ be an inner product space, $T: V \to V$ a linear map, and $A=M(T,P,P)$ for an orthonormal basis $P=[v_1,...,v_n]$ (where $M(T,P,Q)$ is the matrix representation of $T$ with respect to $P$ and $...
1
vote
1answer
44 views

Differential Equation by Minimization

Suppose we want to solve $u + xu' = 0$, which has the general solution $u = \frac{C}{x}$, by minimizing the length squared of $u + xu'$. This should work due to the positive definite condition of ...
3
votes
4answers
125 views

For $\langle Tx, x \rangle \geq \|x\|^2$, prove a solution exists for $Tx = y$.

Edit I've posted this a couple other times, so I now plan on deleting those, and just using this one. Here's the original problem: $H$ is a real Hilbert space. Let $T: H\longrightarrow H$ be a ...
2
votes
1answer
55 views

Are there trilinear inner products?

Is there such a thing as a "trilinear inner product"? The definition of an inner product is: Let $H$ be a vector space over $\mathbb{K}\in \{\mathbb{R,C}\}$. An inner product is a map $\langle \...
1
vote
2answers
70 views

Inner Product of two functions

The inner product of two vectors $\vec a$ and $\vec b$ of n dimensions, is given by, $$(\vec a \,, \vec{b}) =a_1b_1 + a_2b_2 + a_3b_3 + \,\,...\,\,+a_nb_n $$ If a function is considered to be a ...
0
votes
1answer
40 views

Which one of the following option satisfies (1) and (2)?

Consider the subspaces $W_1$ and $W_2$ of $\mathbb R^3$ given by $W_1=\{(x,y,z)\in \mathbb R^3: x+y+z=0\}$ and $W_2=\{(x,y,z)\in \mathbb R^3:x-y+z=0\}$. If $W$ is a subspace $\mathbb R^3$ such that $...