# Tagged Questions

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### Showing the conjugate symmetric property of an inner product when we don't know if our field is $\mathbb{C}$.

The conjugate symmetric property of an inner product states that $\langle{x, y}\rangle = \overline{\langle{y, x}\rangle}$. My question is regarding showing this when we don't necessarily know that our ...
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### Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
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### Few basic things unclear to me about inner product spaces and orthonormal basis

Few things unclear to me about inner product spaces: assume V is an inner product space with B orthonormal basis. Why is it true that: $$\langle x,y\rangle = \langle[x]_{B} , [y]_B \rangle{st}$$ ...
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### Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
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### 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 ...
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### 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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### 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 ...
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### Prove that set is orthonormal set

In vector space of all real polynomials with inner product $(x,y) = \sum_0^1x(t)y(t)dt$. $x_n(t) = t^n$ for $n = 0, 1, \dots$. Show that functions: $y_0(t) = 1,$ $y_1(t) = \sqrt(3)(2t-1),$ ...
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### Why orthogonal basis?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that ...
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### Hoffman & Kunze exercise

This is a problem from Hoffman & Kunze book on linear algebra: Let V be a finite dimensional inner-product vector space. Let U be a self-adjoint unitary linear operator over V. Show that ...
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### How do you prove that tr(B^(T) A ) is a inner product?

Consider the vectorspace of all real $m \times n$ vectors and define an inner product $\langle A,B\rangle = \operatorname{tr}(B^T A)$. "tr" stands for "trace" which is the sum of the ...
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### A basic question on orthogonal vector

Let $V$ be a finite dimensional vector space and $X$ be a subspace. Let $$\langle u,y\rangle=0 \forall u$$ with the property that $$\langle u,x\rangle =0 \;\forall x \in X$$ where $u,y \in V$. Then ...
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### Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
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### find scalar product of vectors in rectangular

let us consider following problem and picture we have $ABCD$ rectagular with $AB=3$ and $BC=5$,$F$ and $E$ are midpoints of rectangular sides,we should find scalar product of my question is ...
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### Prove an equality between dimensions of kernels

Let $V$ be a inner product space over field $\mathbb{R}$ with $\dim(V)<\infty$, and $T\in \text{Hom}(V,V)$. I'm trying to prove:$$\dim(\ker T)=\dim(\ker T^*)=\dim(\ker TT^*)$$ Also, as a conclusion ...
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### A simple piece of a Lemma on Gram-Schmidt

I was looking at a proof of Gram Schmidt theorem and I saw the following lemma, it starts here: First the theorem: if $V$ is an inner product space and $X= \{x_1,\dots, x_n\}$ is a linearly ...
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### In an inner product space over $\mathbb R$, prove $(u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \|$

Let $V$ be an inner product space over field $F$ and $u,w\in V$. Prove that if $F=\mathbb{R}$ then: $$(u,w)=0 \Leftrightarrow \left \| u+w \right \|=\left \| u-w \right \|$$ Is it also true for ...
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### polar decomposition on finite dimensional vector spaces

Let $V$ be a finite dimensional inner product space on $\mathbb{F}$ (where $\mathbb{F}$ can be either $\mathbb{R}$ or $\mathbb{C}$) Let $A$ be a linear operator on $V$. The polar value decomposition ...
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### Dual space and inner/scalar product space

$V$ is vector space of finite dimension. $〈· , ·〉$ is an inner product on $V$.(Field $F$) We set transformation $T \colon V \rightarrow V^*$ as the following: $(T(v))(w) = 〈v , w〉$. Prove that $T$ ...
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### Closed linear subset of a Hilbert space

If $H$ is a Hilbert space, and if $$(a,b)_H=0$$ for every $b \in B \subset H$, where $B$ is a closed linear subset of $H$, does it follow that $a=0$, the zero element of $H$?
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### Revisted: What does $R(T^*)^{\perp}$ mean?

A PLACE WHERE I ADD MY THOUGHTS AS I GO: $$w\in R(T^H)^{\perp}:=\{v\in V ~:~ \langle v, T^Hv\rangle =0~\forall~w\in R(T^H)\}$$ $$\langle v , T^Hv \rangle = \langle ? , ? \rangle =0$$ ...
### $\operatorname{rank} (T^*) = \operatorname{rank} (T)$ : PROOF
Let $T$ be a linear operator on a finite dimensional inner product space. Prove that $\operatorname{rank}(T^*) = \operatorname{rank}(T).$
Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and \$( T^{-1} )^{\star} = ( ...