Consider a Hilbert space which is infinite dimensional. If it is separable, it is well known that an orthonormal basis will be countable, while a hamel basis will be uncountable (since it is a complete space), therefore the two bases cannot coincide. But how can we prove that those two bases cannot be equal if […]

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal basis. I understand the result, but not his calculation. He shows that the inner product of two different exponentials $(e_n (t) = […]

(This is related to this question) $Q \in \mathbb{R}^{n\times k}$ is a random matrix where $k<n$ and the columns of $Q$ are orthogonal (i.e. $Q^T Q = I$). To examine $E(QQ^T)$, I conducted monte carlo simulations (using matlab): [Q R] = qr(randn(n,k),0); In other words, I just sampled a $\mathbb{R}^{n\times k}$ matrix from a standard […]

I am so confused on what to do for this question. The questions asks to find an orthonormal basis of $P_2$, the space of quadratic polynomials, with respect to the inner product $$ \langle p, q\rangle = 2\int_{0}^{1} p(x)q(x)\, dx. $$ I don’t know how to this the question with integrals. All I know is […]

The following is the definition of orthonormal base that I am using: The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an orthonormal basis is a family $\{e_{k}\}_{k} ∈ B$ of elements of $H$ satisfying the conditions: Orthogonality: $\langle e_{k}, e_{j}\rangle = […]

Find an orthonormal basis for the subspace of $\mathbb{R}^4$ that consists of vectors perpendicular to $u = (1, -1, -1, 1)$. I know the components of the vector $u$ is $u_1 = 1, u_2 = -1, u_3 = -1, u_4 = 1$. I managed to find a vector $v$ that is perpendicular to the $u$. […]

I’m trying to randomly generate coordinate transformations for a fitting routine I’m writing in python. I want to rotate my data (a bunch of $(x,y,z)$ coordinates) about the origin, ideally using a bunch of randomly generated normal vectors ($(a,b,c)$ for a plane given by $ax+by+cz+d=0$). The goal is to shift each plane I’ve defined so […]

Suppose $(e_1,e_2,…,e_n)$ is an orthonormal basis of the inner product space $V$ and $v_1,v_2,…,v_n$ are vectors of $V$ such that $$||e_j-v_j||< \frac{1}{\sqrt{n}}$$ for each $j \in \left\{1,2,…,n \right \}$. Prove that $(v_1,v_2,…,v_n)$ is a basis of $V$. I am completely lost and just starting to learn about inner product spaces. Could someone provide a proof […]

I need to show that $v_1,…,v_n$ is basis for $V$ whenever $e_1,…,e_n$ is an orthonormal basis for V and $v_1,…,v_n$ are vectors in $V$ such that $$\left\Vert e_i-v_i\right\Vert < \frac{1}{\sqrt{n}}.$$ I know already that I only need to prove that they are lineary independent. Also I tried to prove it by contradiction, but I got […]

I would like to better understand the gram-schmidt process. The statement of the theorem in my textbook is the following: \noindent The Gram-Schmidt sequence $[u_1, u_2,\ldots]$ has the property that $\{u_1, u_2,\ldots, u_n\}$ is an orthonormal base for the linear span of $\{x_1, x_2, \ldots, x_k\}$ for $k\geq 1$. The formula for $\{u_1, u_2,\ldots, u_n\}$ […]

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