# Tagged Questions

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### Extension from $\Bbb{R}$ to $\Bbb{C}$

i want to prove the following Lemma: Let $\mathcal{H}$ be an $n$-dimensional complex Hilbertspace, $H_1$ its unit sphere and $p:H_1\rightarrow[0,1]$ a proability distribution. Assume that for every ...
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### how inner products are defined on a vector space?

How do mathematicians define inner product on a vector space. For example: $a = (x_1,x_2)$ & $b =(y_1,y_2)$ in $\mathbb{R}^2.$ Define $\langle a,b\rangle= x_1y_1-x_2y_1-x_1y_2+4x_2y_2$. ...
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### Does $S^\bot+T^\bot = (S\cap T)^\bot$ hold in infinite-dimensional spaces?

If $S$ and $T$ are subspaces of some finite-dimensional inner product space then $$S^\bot+T^\bot = (S\cap T)^\bot.$$ See, for example, this post or this post Does it hold in infinite-dimensional ...
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### Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp$

Let $W_1$ and $W_2$ be subspaces of a finite dimensional inner product space space. Prove that $$(W_1 \cap W_2)^\perp=W_1^\perp + W_2^\perp$$ My Try One direction is easy : Let $\alpha \neq 0$ ...
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### Show that $(Au,Bv)=(u,A^tBv)$

Let $A, B$ be matrices of order $n$, and $\vec{u}, \vec{v}$ vectors from euclidean space $\mathbb{R}^n$, then $(Au,Bv) = (u,A^tBv)$ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
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### Question about existence of unitary matrices with certain properties

We are given a set of $d$ normalized vectors on a $d$-dimensional complex vector space: $e_1$, $e_2$... $e_d$, where $$\langle e_j,e_j\rangle=1$$ for all $j$. These are not necessarily mutually ...
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### How can I prove that the span of an a subspace and it's orthogonal complement is the whole vector space?

The book Linear and Geometric Algebra explains the following theorem in a way that I haven't been able to figure out: If $\mathbf{A}$ and $\mathbf{B}$ are subspaces of a vector space $\mathbf{B}$ ...
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### Find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace spanned by $g(x)=x−12$ and $h(x)=1$.

Use the inner product $\langle f,g\rangle =\int_0^1 f(x)g(x)dx$ in the vector space $C^0[0,1]$ to find the orthogonal projection of $f(x)=4x^2−4$ onto the subspace $V$ spanned by $g(x)=x−1/2$ and ...
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### How to find orthonormal basis for inner product space?

In $\mathbb{R}^3$ we declare an inner product as follows: $\langle v,u \rangle \:=\:v^t\begin{pmatrix}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{pmatrix}u$ How can I find an ...
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### Defining an inner abstract vector space

Since an inner product space is an abstract vector space with an additional structure called an inner product, and this additional structure is a component wise operation that associates each pair of ...
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### Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
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### Why $\langle a,x\rangle = \langle b,x\rangle,\forall x\in X\implies a=b$ [closed]

Let $X$ be (possibly infinite-dimensional) Hilbert space. How can we show that if $$\langle a,x\rangle = \langle b,x\rangle,\forall x\in X$$ then $a=b$?
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### Rigid motion on $\mathbb{R}^2$ which fixes the origin is linear

Let $V=\mathbb{R}^2$ be an inner product space with the standard inner product, and let $T$ be a rigid motion of $V$. Suppose $T(0)=0$, prove that $T$ is linear. (A rigid motion of an inner product ...
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### Area preserving transformation in a higher dimensional space is unitary.

In $\mathbb{R}^3$, a linear operator $Q:\mathbb{R}^3 \to \mathbb{R}^3$ preserves the area of parallelograms: that is, given $x,y\in \mathbb{R}^3$, the area of a parallelogram formed by $x$ and $y$ is ...
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### Show that $\langle x, Ax \rangle + \langle b, x \rangle = c$ can be transformed to $\langle x', Ax' \rangle = 1$

Let $A$ be a real, regular, symmetric $n \times n$ matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$ How can I show that $$\langle x, Ax \rangle + \langle b, x \rangle = c$$ can be transformed by ...
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### Transformation of a sum of dot products

I'm not quite sure about this, So I'd like if someone could help me. Can someone explain to me how they get from 2 to 3? \begin{align}\left<2u,u+v\right> &= 0\tag1\\ ...
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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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### Why is the Cauchy-Schwarz inequality considered to be so important?

I've read in the book "Linear Algebra done right" by Axler that the Cauchy-Schwarz inequality is one of the most important results in mathematics. However, in what the book covers and what we have ...
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### Given a symmetric matrix A, find an orthogonal matrix S such that $S^TAS$ is a diagonal matrix

Given the symmetric matrix: $$A = \left( \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{array} \right)$$ find an orthogonal matrix $S$ such that $S^TAS$ is a ...
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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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### Linear algebra: determining if something is an inner product space

If I have a potential inner product space over P2, where $\left< p, q\right> = p(0)q(0)$ How do I determine whether or not it is an inner product space? Using the four axioms I have: ...
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### Determinant (or positive definiteness) of a Hankel matrix

I need to prove that the Hankel matrix given by $a_{ij}=\frac{1}{i+j}$ is positive definite. It turns out that it is a special case of the Cauchy matrices, and the determinant is given by the Cauchy ...
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### 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 ...
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### 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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### 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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### 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 ...
### 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 ...