Tagged Questions

An inner product space is a vector space equipped with an inner product. The inner product is a generalisation of the "dot" product often used in vector calculus.

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28 views

Proving the image of inner product map is whole subspace

I'm doing a specimen exam question and they often have typos and missed pieces of necessary information. I think the question I'm doing might be one such example, but am not sure: We're given that ...
4
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1answer
38 views

Is a function $f$ with $f(X)\perp (I-f)(X)$ necessarily linear?

Let $X$ be a real or complex inner-product space, and let $f : X\rightarrow X$ be a function such that every element of $f(X)$ is orthogonal to every element of $(I-f)(X)$. Prove or give a ...
6
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2answers
56 views

Show that $V = U^\perp \bigoplus U$

If $(V,\langle , \rangle)$ is a Euclidean vector space, $U \subseteq V$ is a subspace of V and $U^\perp := \{v \in V | \langle v,u \rangle = 0, \forall u \in U\}$. Show $V = U^\perp \bigoplus U$ In ...
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1answer
59 views

Orthogonal projection in Inner product space

Let V be $n$-Dimensional ($n\ge1$) inner product space . Let $T:V \rightarrow V$ be a linear map which maintains $ T^2=T$ , $\forall v \in V\ ||Tv||\le||v||$. Prove that there is exists a subspace ...
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1answer
81 views

Symmetric matrix over inner product space

I try really hard to prove this Question. let $A_{nXn}(\mathbb{R})$ Symmetric matrix $A=A^t$ let $\lambda$ be the greatest Eigenvalue of A. we will define over the field $\mathbb{R}$ with the ...
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1answer
44 views

Sufficient condition for two operators being identical on Hilbert space

Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space. If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do ...
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3answers
65 views

Consider a set $S$ of unit vectors in $\mathbb R^2$ such that $\left<x,y\right>=-\frac12$ if $x,y\in S,x\ne y$.

This is a question from an entrance exam paper. Consider a set $S$ of unit vectors in $\mathbb R^2$ such that $\left<x,y\right>=-\frac12$ if $x,y\in S,x\ne y$.Then it is necessarily true that ...
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1answer
129 views

Apollonius’ Identity inner product space

$||z-x||^2+||z-y||^2=\frac{1}{2}||x-y||^2+2||z-\frac{x+y}{2}||^2$ I proved it by expanding both sides and i found both sides are equal. Are there any easy way to prove it?
3
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1answer
93 views

An inequality for inner product space: $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$

In a inner product space show that the following inequality holds. $\|x-z\|.\|y-t\|\leq \|x-y\|.\|z-t\|+\|y-z\|.\|x-t\|$ I am stuck in proving this inequality
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0answers
49 views

Showing an inner product space is complete

I'm working through Ward Cheneys Analysis for Applications and I'm a bit stuck on this exercise from Section 2.2: Prove that if $M=M^{\perp\perp}$ for every closed linear subspace $M$ in an inner ...
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1answer
64 views

Relationship between matrix 2-norm and orthogonal basis of eigenvectors

Given the following matrix: $$ A = \left( \begin{array}{cc} 3 & 4 \\ 0 & 5 \\ \end{array} \right)$$ calculate $\|A\|_2$, with $\|A\|_2 = max_{x \in \mathbb{R}^2 -\{0\}} \frac{\langle Ax,Ax ...
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1answer
57 views

Dot product in Curvilinear Coordinate Systems

I came across the dot product in polar, cylindrical, and spherical coordinates, today. After checking they were equivalent to the Cartesian versions, I started wondering how one would figure them out ...
2
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2answers
25 views

Calculate the angles between $(1,X),(X,X^2),(X^2,X^3),(X^3,X^4)$ given the inner product $\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$

Let $V_4$ be the vector space of all polynomials of degree less than or equal to 4 with the inner product $$\langle p(x),q(X) \rangle = \int_{-1}^{1} p(X)q(X)dX$$ calculate the angles between ...
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1answer
32 views

Let $\langle x,y \rangle = x_1y_1 + 3x_2y_2 + 4x_3y_3 + x_1y_2 + x_2y_1 + x_1y_3 + x_3y_1 + x_2 y_3+x_3y_2$, prove pos. definiteness

Let $$\langle x,y \rangle = x_1y_1 + 3x_2y_2 + 4x_3y_3 + x_1y_2 + x_2y_1 + x_1y_3 + x_3y_1 + x_2 y_3+x_3y_2$$, prove that $\langle x,y \rangle$ is positive definite. I have simplified this to the ...
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2answers
34 views

Norm and InnerProduct Inequality

How can I show that this is true: Let $u,v \in \mathbb{R}^n$: \begin{align} \frac{\|u\|}{\|v\|} \leq \frac{(u,u-v)}{(v,u-v)}, \quad \hbox{if} \quad (v,u-v) > 0 \end{align} Where $\|\cdot\|$ is ...
4
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1answer
37 views

Using inner product property to determine if operator is an isomorphism.

Let $\varphi$ be an operator on a $k$-vector space $V$ with an inner product $\langle\cdot,\cdot\rangle$. Suppose that $\langle v,\varphi v\rangle = 0$ for every $v\in V$. If we take $k=\mathbb R$, is ...
0
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0answers
18 views

Let $G_1 = \{v_1 + \lambda w_1 | \lambda \in \mathbb{R}\}, G_2 = \{\{v_2 + \mu w_2 \}$ be two skew lines, derive a formula for $d(G_1,G_2)$

Let $G_1 = \{v_1 + \lambda w_1 | \lambda \in \mathbb{R}\} \subseteq \mathbb{R}^n, G_2 = \{\{v_2 + \mu w_2 |\mu \in \mathbb{R}\}\subseteq \mathbb{R}^n$, with $v_1, v_2, w_1, w_2 \in \mathbb{R}$ be two ...
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1answer
22 views

self adjoint transformations and inner product problem

Let $(V, <,>)$ be a finite dimensional vector space with an inner product, and let $f,g \in End(V)$ two self adjoint linear transformations. (a) Prove that if $f$ and $g$ commute, then, for ...
3
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1answer
49 views

Angle between two polynomials

Given the inner product of two polynomials $p(X), q(X) \in P(d)$, where $P(d)$ is the vector space of all polynomials of degree less than or equal to d, with real coefficients, and using the inner ...
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1answer
45 views

Isomorphism of inner product spaces

I want to use this in a proof, however I don't know how to prove it itself. I feel as though it's easy to prove by definition but I'm not quite sure.. A linear map V→W between two finite dimensional ...
2
votes
2answers
69 views

Inner product and linear transformation

Let V be an inner product space over $\mathbb{R}$ with inner product ⟨ , ⟩. Let $L:V\rightarrow\mathbb{R}$ be a linear transformation. Show that there is a $\vec{u}\in{V}$ such that $L(\vec{x}) = ...
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1answer
58 views

Evaluating $\int_{\mathbb R^n}e^{-\langle Ax,x \rangle}dx$ where $A$ is symmetric and positive definite [duplicate]

I was asked the following question, and while I think I made little progress, I'd like a push in the right direction. Let $A$ be $n$x$n$ positive definite symmetric matrix with real entries. then for ...
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1answer
40 views

$A \in M_2(\mathbb{R}^2)$,$\langle x,y \rangle = x^TAy$ is an inner product iff $\alpha > 0, det(A) > 0$

Show that given $A=\left( \begin{array}{cc}\alpha &\beta\\\beta&\delta\\\end{array}\right) \in M_2(\mathbb{R}^2)$, $\langle x,y \rangle = x^TAy, (x,y \in \mathbb{R}^2)$, defines an inner ...
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2answers
83 views

Angle between two vectors on manifold

I'm parallel transporting a vector along a curve and trying to calculate how much this vector rotates relative to the curve's tangent vector. So if the path is a geodesic then I will get an answer of ...
2
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1answer
47 views

Inner product definition, definite positive

I'm reading Hoffman and Kunze's linear algebra book and I'm a bit stuck with it's definition of inner product. One of the properties of the inner product is said to be that the product of any vector ...
3
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2answers
74 views

Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
12
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1answer
248 views

Inner product on $C(\mathbb R)$

With Axiom of choice it is possible to construct an inner product on $C(\mathbb R)$. My question is, is it possible to explicitly construct an inner product on $C(\mathbb R)$? I.e. to give a closed ...
1
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1answer
88 views

How to prove $ |\langle u,v\rangle| \leq ||u||||v||$

How to prove $ |\langle u,v\rangle | \leq ||u||||v||$ Note: I have given this many attempts so don't downvote due to lack of effort, refer to edit history for evidence of said effort
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3answers
193 views

Subspace of a finite dimensional inner product space, independence of basis choice

Let $W$ denote a subspace of a finite dimensional inner product space $V$, and let $$\beta = \{w_1,w_2,\dots,w_r\}$$ denote an orthogonal basis for $W$. For any $v\in V$ define $$proj_{\beta}v = ...
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3answers
71 views

Inner Product Spaces, suggestion for book.

Can you suggest me name of some books which would help me visualize IPS better? Like, books having diagrams and stuff?
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3answers
61 views

Difference between Euclidean space and inner product space?

Is it that Inner product space can have infinite dimensions?
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2answers
53 views

Which is the Hermitian inner product, in terms of conjugate and transpose?

Page 29 of Source 1: Denote the complex conjugate by * : $\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$ Page 1 of Source 2: $\mathbf{u \cdot v} = ...
1
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1answer
84 views

Example of $\langle T(v),v \rangle =0$ for every $v$, but $T \neq 0$

As it says in the title, I neeed an example of a linear transformation $T$, over $\mathbb{R}$ (not $\mathbb{C}$!), that satisfies $\langle T(v),v \rangle =0$ for every $v$ in $V$, but $T \neq 0$. Any ...
0
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0answers
26 views

System of equations involving the inner product

I've been reading Ward Cheney's Analysis for Applied Mathematics and he gives the following problem: Indicate how the equation $Ax=b$ can be solved if the operator $A$ is defined by ...
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0answers
30 views

inner product question <,>

Pleace help, that is true or false Let, $a$, $u$ and $v$ $\in \mathbb{R} $, with $v \neq 0 $, $<,>$ is inner product \begin{eqnarray*} <u,v >&=&<a,v>\\ ...
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1answer
14 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
2
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2answers
49 views

Pythagoream theorem

I am preparing a presentation about inner product spaces and I am a little bit confused. I hope anyone can help me to summarize the necessary assumptions for the following theorems: Pythagorean ...
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2answers
38 views

Scalars determine a vector in inner product space.

Let $V$ be a finite dimensional inner product space over $k$ with basis $\{v_1,\dots,v_n\}$ and inner product $\langle \cdot,\cdot\rangle$. For any $\alpha_1,\dots,\alpha_n\in k$, there exists a ...
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2answers
69 views

Why is there an “absolute value” and a norm in the Schwarz Inequality?

This really bothers me, and I'm not sure if it's just that I'm not understanding it correctly. For the moment, assume we are working in a vector space $V$ over $\mathbb{R}^n$. Let $x,y \in V$. We have ...
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0answers
35 views

Complex Gram-Schmidt

Consider a complex vector space where we have $n$ linearly independent vectors and we want to create an orthonormal basis with Gram-Schmidt algorithm. I'm going to denote the original vectors with ...
2
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0answers
20 views

Using derivatives at 0 to define an inner product

Can the following define an inner product on a subspace of the set of functions that are infinitely differentiable on $[-R,R]$. If so, do we get a Hilbert space? $$<f, g> = \sum_{n=0}^\infty ...
0
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2answers
45 views

Equations involving the inner product

I've been reading chapter 2 of Ward Cheney's Analysis for Applied Mathematics, and he gives the following question: Find all solutions to the equation $\langle x,a \rangle c = b $, assuming that ...
0
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2answers
44 views

Why are these vectors expressed as row vectors and not column vectors? When to write as row vectors or column vectors?

Everytime I have been asked to find a basis when the vectors were given in comma delimited form, I, and the book, would write out the vectors as columns in a matrix. Another example in the book ...
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2answers
20 views

Basic question on inner product and norms

If $x = (f,g)$ is a vector in $\mathbb{C}^2$. such that $\|x\|^2 = |f|^2 + |g|^2 = 1$ Define the operator $T: \mathbb{C}^2 \to\mathbb{C}^2$ as $Tx = (g,0)$, then $\langle Tx, x\rangle = g\bar{f}$. ...
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1answer
21 views

Finding the orthogonal complement

Here's the question $$ V = \{ f: [0,1] \to \mathbb{R} \mid f \text{ continuous} \}. $$ Find the orthogonal complement for $$ U = \{f \in V \mid \forall x \in [0,0.5], f(x)=0\} $$ (with the ...
2
votes
0answers
31 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
0
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1answer
34 views

Prove: given a basis of vector space, we can find an inner product such that this basis is orthonormal

$V$ is vector space above fields $\mathbb{R}$ or $\mathbb{C}$ , and $B = \{v_1,...,v_n\}$ is a basis of $V$. I need to prove that there is an inner product on vector space $V$, such that $B$ is an ...
0
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0answers
30 views

Custom Norm Function Proof $\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} \right | $

For Vector Space X consisting of ordered pairs of Complex numbers, Can we define the Norm stated below from inner product, in X ? $$\left \| x \right \|=\left | \xi _{1} \right |+\left | \xi _{2} ...
5
votes
1answer
103 views

Is it possible to define an inner product such that an arbitrary operator is self adjoint?

Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$. The definition as stated require us to start with an ...
1
vote
1answer
16 views

conditions for positivity invariance in inner products

Let $R = R^{\top} \in \mathbb{R}^{m \times m}$ be a positive definite matrix. My question is: Under which conditions $u^{\top} R w > 0$ implies $u^{\top} w > 0$ ? Rejecting the trivial case ...