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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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? ...
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If $Ax = O$ has only one solutions, then the columns of A: ${v1, v2…,vn}$ span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
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Adjoint of linear transformation $T: \mathbb{M_n(C)} \rightarrow \mathbb{M_n(C)}$

Let V = ${M_n(\mathbb C)}$ with inner product $\langle A, B\rangle = \text{tr}\,(B^*A)$, $A, B \in V$. Let $M \in {M_n(\mathbb C)}$, Define $T: V \rightarrow V$ by $T(A) = MA$. What is adjoint of ...
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Showing that $Im(L^*)=(Ker\: L)^\perp \space \:\mathrm and \:\:Ker(L^*)=(Im\: L)^\perp$

Let $V,W$ be finite-dimensional euclidean or unitarian Spaces and $L: V \to W$ a linear map. I have to show the following: $$Im(L^*)=(Ker\: L)^\perp \space \:\mathrm {and} \:\:Ker(L^*)=(Im\: L)^\perp ...
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family of scalar products

Does there exist an uncountable family of inner products defined on some vector space such that any two norms induced by these products are not equivalent?
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Finding a maximal orthogonal basis from a set of functions

I have a set of functions $\{f_1,\ldots,f_n\}$ with an associated inner product $\langle f_j,f_k\rangle=\int d^2z f_1^*f_2$ . The functions are not linearly dependent; i.e. the rank $r<n$, where ...
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Some property of a sequence in Hilbert space [duplicate]

Let $y_1, y_2, \cdots$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, \cdots, y_n\}.$ Assume that $||y_{n+1}|| \leq || y -y_{n+1}||$ for each $y \in V_n$ for $n = 1, 2, ...
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Positive definiteness of a bilinear form implies symmetry?

In the Wikipeda article about positive definite bilinear forms, there is the line It turns out that the matrix $M$ is positive definite if and only if it is symmetric and its quadratic form is a ...
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Inner product of continuous functions

I'm sure i'm missing something but i've come to this definition of inner product: $$\langle f, g \rangle = \int_a^b f(x)g(x) \ dx$$ For functions $F:[a,b]\rightarrow \mathbb{R}$ Now, I know that ...
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How to calculate $\langle v,w\rangle$ based only on $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$?

Let's assume $v,w, x_i \in R^n$ are unknown. Can one compute dot product $\langle v,w\rangle$ if one has just the numbers: $\langle v,x_i\rangle$ and $\langle w,x_i\rangle$ for $n$ random vectors ...
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$\langle x+y,x-y\rangle=0$

$(X,\langle.,.\rangle)$ is an inner product space over $\mathbb{R}$. If $x, y \in X$ where the induced norms of $x$ and $y$ are equal, prove that $x+y$ and $x-y$ are orthogonal. So I want to show ...
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Definition of inner product space

In the definition, we defined linearity in the first argument, Hermitian symmetry. And these two imply anti-linearity in second argument. Is it equivalent, if I cancel the Hermitian symmetry and only ...
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Finding the orbits of the orthogonal group $O(n)$ on $\Bbb R^n$

Let $O(n)=\{M\in GL_n(\mathbb{R}):MM^t=M^tM=I\}$ an orthogonal group. I need please an explain why each orbits consists of all vectors with the same length. I know that an orbit is defined by ...
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If $(U,〈\;⋅\;,\;⋅\;〉),H$ are Hilbert spaces, $W\in U$, $Y\in H$, $Z\in L(U, H)$ and $f\in L(H,L(H,\mathbb R))$, then $〈Y,fZW〉=〈ZW,fY〉$

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be Hilbert spaces $W\in U$, $Y\in H$ and $Z\in\mathfrak L(U, H)$$^1$ $f\in\mathfrak L\left(H,\mathfrak L\left(H,\mathbb R\right)\right)$ How ...
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Adjoint Operator to the Derivative

Let $V \subset \Bbb R[X]$ be the Vectorspace of all Polynomials of degree $\le 3$. The inner product on $V$ is defined as follows: $$\langle f,g \rangle:=\int^1_{-1}f(t)g(t)dt$$Let $L:V \to V$ be the ...
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Show that $T$ is unitary if and only if we have an orthonormal basis

Let $\{v_1,...,v_n\} \subset V$ be an orthonormal basis of $V$. Show that the set $\{T(v_1),\dots,T(v_n)\}$ is an orthonormal basis of $V$ if and only if $T$ is unitary. I have probably gone into ...
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Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
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Prob. 6, Sec. 3.4 in Kreyszig's functional analysis book: The fourier coefficients minimise the distance.

Let $n \in \mathbb{N}$, let $\{ e_1, \ldots, e_n \}$ be an orthonormal set in an inner product space $X$, let $x \in X$, let $y(x) \colon= \sum_{j=1}^n \langle x, e_j \rangle e_j$, and let $z \colon= ...
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orthogonal projections on an IP space proof

Let $E_1,E_2$ be orthogonal projection on an inner product space V. Prove that $E_1E_2=0$ if and only if the range subspaces of $E_1$ and $E_2$ are orthogonal. I can prove one direction pretty ...
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Show that $Ax \geq 0$ implies $\langle b,x\rangle \geq 0$ for $x \in \mathbb{R}^n$

I came across this questions in one of my optional courses. I am trying to apply separation theorem but not very sure how to proceed Suppose there are two dates, 0 and 1. Suppose the world will be in ...
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1answer
9 views

How to find basis for orthoogonal complement basis for the following condition?

Let $W=$span$\{\begin{bmatrix}1&1\\0&0\end{bmatrix}\begin{bmatrix}0&0\\1&1\end{bmatrix}\}$ and suppose the span is orthogonal under certain Hermitian inner product space (just ...
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Miscellaneous problem (M.12) from M. Artin's Algebra 2nd edition Chapter 8.

Let $A$ be a real orthogonal $n\times n$ matrix with $X$ complex eigenvector with complex eigenvalue $\lambda$. Prove that $X^tX=0$. Then if $X=R+Si$ where $R$ and $S$ are real vectors. Prove that ...
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Show that $u$ is a multiple of $v$ or vice versa in the following

Show that if $\left | \left \langle u,v \right \rangle \right | = \left\|u \right \| \left\|v \right \|$ then either $u$ is a multiple of $v$ or $v$ is a multiple of $u$. Here is what I did: Assume ...
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If $T$ is a self-adjoint linear operator on an inner product space $V$ such that $T\circ T=T$ then $T$ is projection on $Image(P)$.

Let $T$ be a self-adjoint linear operator on an inner product space $V$ such that $T\circ T=T$. Let $W=Image\:T$ Prove that for all $v\in V$, $T(v)=proj_W(v)$ ie $T$ projects $v$ onto it's image. Now ...
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Are there some non-complete inner product spaces in which the equality holds:$M^{\perp\perp}=\overline{\operatorname{span} M}?$

Let $X$ be a Hilbert space and $M\subset X$. We know that the following is true: $$(M^{\perp})^{\perp}=\overline{\operatorname{span} M}.$$ But I want to know is it true if $X$ is an inner product ...
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Is the following Hermitian inner product space projection correct?

Suppose we have orthogonal basis $v_1=\begin{bmatrix}i\\1\\0\end{bmatrix}, v_2=\begin{bmatrix}-i\\1\\2\end{bmatrix}$. We want to find proj$_W(\begin{bmatrix}2-i\\1\\i\end{bmatrix})$. Let ...
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How to use Gram-schmidt process on Hermitian inner product space?

So $W=$ span$\left\{\begin{bmatrix}i\\1\\0\end{bmatrix},\begin{bmatrix}-1\\1+i\\1\end{bmatrix}\right\}$ is a subspace under complex space $\Bbb{C}^3$. To get an orthogonal basis, we use the ...
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Does this inner product manipulation make sense?

Suppose A is a normal matrix over $M_n(\mathbb{C})$, with diagonalization $A = PDP^*$. Consider the inner product $<A\mathbf{v}, \mathbf{v}>$ $<A\mathbf{v}, \mathbf{v}> = ...
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find a scalar $\alpha$ that satisfy the following

when $v\neq 0$, find a scalar $\alpha$ such that $z:=u-\alpha v$ satisfies $\left \langle z,v \right \rangle = 0$ Is there some trick to this? I tried solving this explicitly and I just ended up ...
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Are linearly independent harmonic polynomials orthogonal upon integration over the sphere?

There is a theorem that states that the vector space of homogeneous polynomials decomposes into an orthogonal direct sum of vector spaces of harmonic polynomials as $$ \mathcal{P}_n = \mathcal{H}_n ...
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Show $\exists u,v \in X$ such that $\|u+v\|^2 \neq \|u\|^2+\|v\|^2$

Let $(X, \langle \cdot , \cdot \rangle )$ be an inner product space. Show that if $X \neq {0}$ then there exist $u,v∈X$ such that $\|u+v\|^2 \neq \|u\|^2+\|v\|^2$. Can we just say that choose $u$ ...
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Representing inner products on $\mathbb{R^n}$ using matrices

I would like to ask two basic questions about inner products on $\mathbb{R^n}$: Given an inner product $\langle u, v\rangle$ on $\mathbb{R^n}$, 1) Am I correct in thinking that the matrix $A$ with ...
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Is the dot product not applicable for complex vectors?

Is the concept of dot products not applicable for vectors involving imaginary #s? Are dot products a subset of inner products?
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Verification: $S^{\perp} = ({span(S)})^\perp = (\overline{S})^\perp = (\overline{span(S)})^\perp$?

Let V be an inner product space (not necessarily Hilbert) and S a subset. This question Orthogonal complement of a Hilbert Space deals with $S^{\perp} = (\overline{span(S)})^\perp$. I'm looking for ...
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Proving $\langle f,f \rangle =0 \implies f=0$

Let $Y=C[0,1]$, prove that $\langle f,g \rangle = \int _0^1 f(x)g(x)dt$ is an inner product. Just need to talk about the condition on the title. Can we simply say $<f,f>=0$ so $||f||=0$ so by ...
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Find $g \in f^{\perp}$ s.t. $\langle f,g \rangle =0$

Let $(Y, \langle .,. \rangle )$ be an inner product space with $Y=C[0,1]$ and $$\langle f,g \rangle = \int_0^1 f(x)g(x)dx$$ In $C[0,1]$, let $f(t) =t$, and find $g \in f^{\perp}$ such that ...
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Finding the shortest vector in a subspace of an inner product space

In the inner product space $L^2$[$0,1$], find a shortest vector to $f(x)=sin(πx)$ in the subspace $W=[a_0+a_1x+a_2x^2|a_0,a_1,a_2∈$R$]$. How does one prove that the vector found is a shortest one?
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Converting inner product matrix to an identity matrix

I'm working on a practice exam, and I am having a lot trouble finding the solution to this problem. The solution's are posted, however they seem to be completely computationally wrong. In the hours I ...
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Question about two ways to induce an inner product on $S^2V$

$\newcommand{\til}{\tilde}$ Let $(V,g)$ be an $n$-dimensional inner product space, and let $S^2V^*$ be the symmetric algebra. I am familiar with a natural way to endow $S^2V^*$ with an inner product ...
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Proof of claim on orthonormal elements in an inner product space

Let $X$ be an inner product space and $\{e_{n}\}_{n=1}^{\infty} \subset X$ be an orthonormal set. Show that $$ \sum_{n=1}^{\infty}|\langle x,e_{n}\rangle\langle y, e_{n}\rangle| \leqslant ...
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Inner Product proof, axiom 1

I hate to argue that my text book is wrong. That being said, I am going to try and do just that. The text book says that this IS a valid inner product, I disagree. The vectors u and v are defined ...
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Find $y \in W_{2}^{1}[-1,1]$ s.t. $\forall x \in W_{2}^{1}[-1,1]$, $f(x)=\langle x, y \rangle$

Consider a Sobolev space $W_{2}^{1}[-1,1]$ with the following inner product: $\langle x, y \rangle = \int_{-1}^{1} [x(t)y(t)+x^{\prime}(t)y^{\prime}(t)]dt$. Let $f(x) = \int_{-1}^{1}e^{2t}x(t)dt$. ...
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1answer
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Showing that $\langle T(u),T(v)\rangle = \langle u, v \rangle$ implies $T$ is a linear isometry

Let $T$ belong to $\mathcal{L}(H)$ (i.e., the set of linear operators from $H \mapsto H$ where $H$ is a Hilbert space). I need to show that $T$ is an isometry iff $\langle T(u),T(v) \rangle = \langle ...
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How to find $W^{\perp}$ in the following polynomial inner product space?

Consider $P_3(\Bbb{R})$ with inner product $\langle p(x),q(x)\rangle=\int^1_{-1} p(x)q(x)dx$ and let $W=\{ p(x)\in P_3(\Bbb{R})|p(0)=p'(0)=p''(0)=0\}$. How to find $W^{\perp}$? Let's set ...
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Equality in the Cauchy-Schwarz inequality implies $\overrightarrow v$,$\overrightarrow w$ linearly dependent

Show that one gets equality in the Schwarz inequality if and only if $\overrightarrow v$,$\overrightarrow w$ are linearly dependent. (I am supposing they want me to prove it in an inner product space ...
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$T$ is self-adjoint $\Rightarrow \exists$ positive $A,B$ such that $T=A-B$ and $AB=0$

I have a trouble by the following problem and I dont have any idea to solve it. can anybody give me a hint? Thanx in advance. Let $\mathcal H$ be a Hilbert space and $T:\mathcal H \to \mathcal ...