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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Parallelogram law in Hilbertspaces [on hold]

I am having trouble solving the following exercise: (a) Let $(V, ||\cdot||)$ a normed vector space over the field $\mathbb R$. Show that, for all $x, y \in V$ the equation $$ ||x+y||^2 + ...
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The signature of an inner product space does not depend on its basis

In R.W.R. Darling's "Differential Forms and Connections" an inner product is defined for a vector space $V$ as a bilinear, symmetric and nondegenerate (but not necessarily positive-definite) map from ...
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31 views

Self-adjoint operators and eigenvalues

In a previous exam the following question was asked which I was unable to answer due to the lack of knowledge of self-adjoint operators. Let $S$ be a self-adjoint operator on a real finite ...
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Orthogonal complement of diagonal matrices

My question is similar as this one: Orthogonal complement of the diagonal matrices in the inner product space of matrices Considering $\mathbb{R}^{n\times n}$ with inner product $<A,B> = ...
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24 views

finding inner product

This is from my textbook: I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, ...
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26 views

Prove that the dimension of row space equals to the dimension of column space of an $n\times n$ matrix

Knowing that the row space of $A\in \mathbb{R}^{n\times n}$ equals $N(A)^\perp$ prove that the dimension of column space of a matrix equals its row space dimension. So I'm trying to apply ...
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Determine a projection of a integral

Determine the polinomial function p(t) belonging to $\mathbb{R}_2[t]$ so that the value of $\int_{0}^{2\pi} [p(t)- \cos(t)]^2 dt$ is minimal. So as this is problem of linear algebra ...
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31 views

Finishing a proof with inner product

Consider a linear space with inner product $V$ and vectors $x,y \in V$. Show that if $<v,y> = <v,x>$ for all $v\in V$, then $x=y$. My attempt: So what i basically did was to develop ...
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34 views

Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So ...
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Proof with inner product of vectors

Admit that $V$ is a linear space with inner product. Show that given $x,y \in V$ we have $<x,y>=0$ if and only if for every scalar $\alpha \in \mathbb{K}$ we have $|x|\le |x+ \alpha y |$ My ...
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for every linear map $\ T:V\to V$ : $\ [T^*]_B=(M^t)^{-1}A^tM\ $ when $\ [T]_B=A$.

Let $V$ be an inner product space of finite dimension over $\mathbb{R}$ and Let $B=\{v_1,...v_n\}$ be a basis of V (not necessarily orthonormal). Let $M\in M_n(\mathbb{R})$ a matrix whose i,j ...
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28 views

Difference of inner product space of two vectors

If in an inner product space $\alpha,\beta$ are two vectors such that $\|\alpha\|= 2,\|\beta\|=3$, and $\|\alpha+\beta\|=5$. Then $\|\alpha-\beta\|$ is equal to ? The options are 1)0 2)1 3)√10 ...
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43 views

Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle ...
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35 views

Gram-Schmidt process - Division by zero (ERROR)

I'm working with full-rank lattice basis, and I need to compute the Gram-Schmidt norms and coefficients to measure its quality. But during the process I have a division by 0. The division by zero is ...
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27 views

Inequality between angles with respect to different inner products

Given a finite dimensional real vector space $V$ with two inner products $\langle,\rangle_g$ and $\langle,\rangle_{\tilde{g}}$, what are the inequalities for angles between vectors? Is there a similar ...
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45 views

An expression for the Hilbert-Schmidt inner product

Suppose that $k:[0,1]\times[0,1]\to\mathbb C$ is a Hilbert-Schmidt kernel, i.e. $$ \int_0^1\int_0^1|k(x,y)|^2\mathrm dx\mathrm dy<\infty. $$ The associated Hilbert-Schmidt integral operator ...
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13 views

Homogeneous functions and inner products

I am having trouble with understanding how homogeneous functions are related to the inner product. I'm trying to prove the following: For a functions $f:\mathbb{R}^n\rightarrow\mathbb{R}$, $f$ is ...
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25 views

norm from inner product

I have a question in a Hilbert Spaces course as follows: Let $X=(x_1, x_2)$ be vector in a vector space of all ordered pairs of complex numbers X. Can we obtain the norm defined on X by: ...
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28 views

Two different notations for inner product

In my linear algebra course on the faculty of mathematics, we used the following notation for the inner product: $$\langle v, w \rangle$$ On the other hand, my friends from the faculty of physics ...
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25 views

Characterization of orthogonality in an inner product space.

How can I show the following assertion In an inner product space, $x\perp y$ if and only if $\|x+\alpha y\|\ge \|x\|$ for all scalars $\alpha$.
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67 views

Inner product of a finite sum

I have been working on this question for very long, but I can't find the right answer. I know that I have to check the properties for an inner product, but it gives me a hard time. The question is: ...
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52 views

Getting an isomorphism from a short exact sequence of inner product spaces

Let $L,M,N$ be finite dimensional inner product spaces and $0 \to L \xrightarrow{\alpha} M \xrightarrow{\beta} N \to 0$ is a short exact sequence. Now let $\beta^* : N \to M $ be the adjoint map (the ...
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134 views

Linear Algebra - Preservation of inner product

Consider the vector space $\mathbb{R}^2$ with the standard inner product given by $ \langle(a, b), (c, d)\rangle = ac + bd$. (This is just the dot product.) (a) Let $\theta \in [0,2\pi)$ and let T : ...
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If $T$ is normal, and $M_1, M_2$ are coprime polynomials such that $M_1 (T)u=0=M_2 (T)v$, then $u \perp v$

Let $T$ be a normal transformation in an inner product space of finite dimension $V$. Let $M_1 (t), M_2 (t)$ be coprime polynomials and let $u,v \in V$ be vectors. Prove that if $M_1 (T)u=0$ and ...
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Minimize and maximize the sum of dot products at the same time

this is the problem. I have a set of numerical positive vectors of equal length. For each pair of vectors $(\mathrm{i}, \mathrm{j})$ I define the vector $\mathrm{ij}=\mathrm{i} - \mathrm{j}$. I also ...
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56 views

Euclidean Inner Product in R^n

A matrix defined inner product of R^n generated by the invertible nxn matrix A, < u,v >= Au dot Av. An orthogonal matrix is an invertible matrix where A^T=A^-1 The question asks to prove that if A ...
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Prove $\langle v,v\rangle=\langle w,w\rangle $ implies $v=w$ [closed]

This may seem rather simple, but I'm trying to rigorously confirm this intuition. Can anyone help? Suppose v,w are elements of V. Prove: $\langle v,v\rangle=\langle w,w\rangle\ \implies v=w$.
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34 views

self-adjoint / orthonormal basis of eigenvectors [closed]

Let $T$ be a linear operator on a finite dimensional real inner product space $V$.Then $T$ is self-adjoint iff there exists an orthonormal $\beta$ for $V$ consisting of eigenvectors of $T$. Please ...
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49 views

Frechet derivative in a Hilbert space

Let $\mathcal{H}$ be a Hilbert space and $A$ a self-adjoint operator. With $(\, ,\, )$ denoting the inner product and $\psi\in \mathcal{H}$, I want to formally show that the Frechet derivative of the ...
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Require the understandable proof of Schur's theorem for Linear operators.

Schur's theorem for Linear operators Let $T$ be a linear operator on a finite dimensional inner product space $V$. Suppose that the characteristic polynomial of $T$ splits. Then there exists an ...
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Linear Independence and Inner Product Spaces

Let $S=\{v_1,v_2,\dots,v_n\}$ be a linearly independent subset of an inner product space $V$, and $w \in V$ where $w$ is orthogonal to each vector in $S$. Prove, using only the definition of linear ...
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the inner product of 2 vectors in complex vector space

Let us consider two vectors $u$ and $v$ are in the complex vector space. The inner product of these two vectors is defined by $u\cdot v=(u_1^*\cdot v_1, u_2^*\cdot v_2,\cdots\cdots, u_n^*\cdot v_n)$ ...
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How to find the orthogonal projection of the given vector on the given subspace $W$ of the inner product space $V$.

$V=P(R)$ with the inner product $\langle f(x),g(x) \rangle$=$\int_0^1 f(t)g(t )dt$, $h(x)=4+3x-2x^2$ and $W=P_1(R)={\{1,x}\}$. I don't know how to do this the question. All I know is that it has ...
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Find the orthogonal projection of the given vector on the given subspace $W$ of the inner product space $V$.

$V=\mathbb R^2,u=(2,6), $ and $W={\{(x,y):y=4x}\}$. I've no idea about how to get through this. Please help in understanding this in detail,if possible pictorial representation will be best.
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Find $\langle f,g \rangle$ w.r.t. $L_0 \perp L_1$.

Let $X=C[-1,1]$, and $L_k= \{ <t^{k+2i}, i=0,1,2,... > \} $. Define an inner product on $X$ with respect to $L_0 \perp L_1$. Then confirm that $L_0 \perp L_1 $ on your inner product. Can we ...
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Characterizing when kernels in Reproducing Kernel Hilbert Space (RKHS) are linearly independent

In my studies of RKHS i.e. Reproducing Kernel Hilbert Spaces stating the following Let $ \mathbb{H} $ be a RKHS on a set X. We are asked to characterize when the following set of kernels $ \{ ...
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Let $V$ be an inner product space over $\mathbb{C}$, prove the Polar Identities: For $x,y\in V$

Let $\langle x,y\rangle=\displaystyle\frac{1}{4}\sum\limits_{k=1}^4 i^k\lVert x+i^ky\rVert^2 $ where $F=\mathbb C$ and $i^2=-1$. Proof: We have $$ \begin{align} \lVert x+i^ky\rVert^2 &= ...
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The very definition of an inner product

Let's consider a finite, n-dimensional inner product space. My first question is, is the inner product between a pair of vectors $v_1 = \sum_{i=1}^n \alpha_ie_i$ and $v_2=\sum_{i=1}^n \beta_ie_i$ ...
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45 views

Is $X$ one dimensional

If $X $ is an inner product space and if there exists $x \in X $ s.t. $\{x\}^{\perp}=0$. Is $X$ one dimensional? The way I have written out this question a bit wrong because it was in my exam so I ...
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42 views

Deduce the Pythagorean Theorem [duplicate]

Let V be an inner product space,and suppose that $x$ and $y $ are orthogonal vectors in V.We also know $\left\lVert x+y \right\rVert^2=\left\lVert x \right\rVert^2+\left\lVert y \right\rVert^2$.My ...
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In $\mathbb C^2$, Show that $\langle x,y\rangle=xAy*$ is an inner product.

$A= \left[ \begin{array}{cc} 1&i\\ -i&2 \end{array} \right] $ I've shown that (a) $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$. (b) $\langle ...
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Explicit example of two non-isomorphic Hilbert spaces with the same algebraic dimension [duplicate]

I´m wondering if there exist a vector space A and inner products: $\langle\cdot{,}\cdot\rangle_1$ and $\langle\cdot{,}\cdot\rangle_2$, such that: $\big( A,\langle\cdot{,}\cdot\rangle_1 \big)$ and ...
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Normed space, inner product space, which is larger?

My textbook says normed space is inner product space unless it satisfies some parallelogram identity. In this case, we can conclude inner product space is a subset of normed space. However, norm is ...
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Let V be an inner product space.Then for x,y,z belongs to V and belongs to field,F,the following statements are true.

(a) $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$. (b) $\langle x,cy\rangle =\bar c\langle x,y\rangle$. (c) $\langle x,0\rangle = \langle 0,x\rangle =0$. (d) $\langle x,x\rangle=0$ ...
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Prove that the inner product $\langle f,g\rangle= \int f(t)g(t) \, dt $ satisfies the property positive definiteness [duplicate]

In $C[a,b]$, define the product $\langle f,g\rangle= \int f(t)g(t) \, dt $. Show that this product satisfies the property, $\langle f,f\rangle$ is greater than zero for all non zero $f$ using a ...
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Proof: Sum of dimension of orthogonal complement and vector subspace

Let $V$ be a finite dimensional real vector space with inner product $\langle \, , \rangle$ and let $W$ be a subspace of $V$. The orthogonal complement of $W$ is defined as $$ W^\perp= \left\{ v ...
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Is $\langle A,B\rangle =\operatorname{trace}(AB^T)$ an inner product in $\mathbb R^{n\times m}$?

I don't understand why one should take transpose of $\operatorname{tr}(AB^T)$ and why we use the fact that $\operatorname{tr}(M)=\operatorname{tr}(M^T)$ for any $M$ that is a square matrix to solve ...
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Subspace of Inner product space is an Inner product space

How do I prove that a subspace $ M $ of Inner product space $V$ is an inner product space? Isn't it obvious that the inner product of any two vectors from $M$ satisfies axioms? Does it require a ...
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In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
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multiplication of finite sum (inner product space)

I am having difficulty to understand the first line of the proof of theorem 3.22 below. (taken from a linear analysis book) Why need to be different index, i.e. $m,n$ when multiplying the two sums? ...