# Tagged Questions

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### Gram-Schmidt: Do the sets have some sort of order?

I'm learning about the Gram-Schmidt process: I have some subspace basis $A$ with three vectors: $$A = \{a_1,a_2,a_3\}.$$ Based on it, we will create an orthonormal basis $B$ with three vectors, ...
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### Orthonormals, v1,v2.

Given: $2$ vectors $v1,v2 \ne 0$ and $v1 \ne v2$ and $v1,v2 \in R^n$ Prove: $\{v1\}^\bot = \{v2\}^\bot$ if and only if ${v_1,v_2}$ are linearly dependent. Well, I do have a solution for this but I ...
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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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### Orthogonals,Orthonormals,bases.

Given: $B = ((2/3, 1/3,2/3),(1/3,2/3,-2/3),(2/3,-2/3,-1/3))$ How can I check if B is an orthonormal base of $R^3$? I thought about first checking if these vectors are a base. Then checking if they ...
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### Does R' = R*R'*R^(-1) hold?

Given two 3D rotation matrices ($3\times 3$) $R$ and $R'$, does this equivalence hold?: $R' = R*R'*R^{-1}$ My intuition tells me so, but I can't find a formal proof for it. Thanks.
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### Orthonormal matrix by diagonal matrix by orthonormal matrix transpose. [closed]

Let $D$ be a diagonal $n \times n$ matrix and $V$ an $n\times n$ matrix whose columns are orthonormal. Is it true that $$VDV^T = D?$$ If so, why?
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### Name for multiples of orthogonal matrices

Is there a name for a matrix which is a multiple of an orthogonal matrix? I.e. a square matrix $A$ which satisfies the condition $$A^TA = AA^T = \lambda I$$ where $\lambda$ is some scalar (which ...
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### Orthonormal basis of $\mathbb{R}^n$ containing $v_1 = \frac{1}{\sqrt{n}}(1,1,\ldots,1)$

I would like to construct the orthonormal basis $\{v_1,v_2,\ldots,v_n\}$ of $\mathbb{R}^n$ with $v_1 = \frac{1}{\sqrt{n}}(1,1,\ldots,1)$. I am looking for an analytic formulation of all vectors of the ...
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### small question about gram schmidt

I tried to find an answer online first, so excuse me if this question was asked before, but, is the result of the gram schmidt process different, depending on the order that I go over the vectors in ...
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### What am I doing wrong? Gram Schmidt process..

Let there be the inner product of all polynomials of degree smaller or equal to 2: $\langle f,g\rangle=\int_0^1f(x)g(x)xdx$. Find orthonormal basis. So I really tried this for an hour and it pretty ...
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### Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
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### Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
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### Difference between Householder Reflections and Gram-Schmidt?

In numerical QR decomposition, when we calculate the orthonormal factor Q of a matrix, what is the difference in results if we use Householder Reflections to normalize the matrix or use Gram-Schmidt ...
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### Question about Gram-Schmidt algorithm. Orthogonal diagonalization. Does GS conserve eigen-ness property

I have a question about the Gram-Schmidt process, and about the algorithm to find an orthogonal basis of eigenvectors (aka orthogonal diagonlization). let $T:V \to V$ be a diagonlizable linear ...
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### Orthornomal matrices [duplicate]

Is there a more direct reason for the following: If the columns of $n\times n$ square matrix are orthonormal, then its rows are also orthonormal. The standard proof involves showing that left ...
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### Question about orthogonal transformation / orthogonal matrices

I have a question about orthogonal transformations. If $T$ is an orthogonal transformation from $V$ to $V$, should the representation matrix with respect to any orthonormal basis of any inner product ...
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### $A$ is a symmetric real matrix. Show that there is $B$ such that $B^3=A$

I'm having trouble with this question, I'd like someone to point me in the right direction. let $A$ be a n by n matrix with real values. show that there is another n by n real matrix $B$ such that ...
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### Proving the standard matrix U of T to be orthogonal

So my class is getting into orthogonality, however, our reading assignments haven't been touching on transformations. I have this proof problem that I cannot seem to get around. Does anyone have any ...
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Let $V$ (which is infinitely dimensional) be the set of all continuous functions $\Bbb{S}^1 \to \Bbb{R}$. Show that $V$ is a vector space. Define $\langle-,-\rangle: V\times V\to \Bbb{R}$ by $$\langle ... 1answer 115 views ### matrices in the gram schmidt process I have a question that says: Use the Gram-Schmidt orthonormalization process to transform the given basis for a subspace into an orthonormal basis for the subspace. Be sure to show your matrix ... 1answer 116 views ### How to prove det(A) = 1 or -1 \Longrightarrow AA^t = A^tA = I_n? Prove det(A) = 1 or -1 \Longrightarrow AA^t = A^tA = I_n? I have no clue, to be fair. I am trying to prove orthogonal polynomials have a det = 1 - help? 1answer 91 views ### If \{u,v\} is an orthonormal set in an inner product space, then find \lVert 6u-8v\rVert If \{u,v\} is an orthonormal set in an inner product space, then find \|6u-8v\|. That's pretty much it. I'm trying to study for a quiz and can't figure it out. There are no examples in my book to ... 1answer 58 views ### Orthogonal M such that M^TAM is upper triangular? We triangularise$$A=\begin{bmatrix}2&1&1\\-20&-5&-10\\16&3&8 \end{bmatrix}$$into U=S^{-1}AS with$$S=\begin{bmatrix}-1&0&0\\0&-1&0\\2&1&1 ...
Given a linear operator $T$ on an $n$-dimensional vector space $V$ (over $\mathbb R$), I want to find an orthonormal basis for $V$ in which the matrix of $T$ is sparse (has many zeros). How many zeros ...
I have a question: Find the projection of $v = <1,2,1>$ on $span(<3,1,2>,<1,0,1>)$ in $R^3$ calling the vectors in the span a and b proj_w V = \frac{V \cdot a }{a^2} ...