1
vote
1answer
57 views

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, ...
1
vote
2answers
38 views

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 ...
1
vote
2answers
22 views

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 ...
0
votes
2answers
19 views

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 ...
0
votes
3answers
58 views

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.
0
votes
1answer
32 views

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?
0
votes
2answers
29 views

Orthonormal bases with canonical basis

Consider $\mathbb{R}^3$ with two orthonormal bases: the canonical basis $e = (e_1,e_2,e_3)$ and the basis $f = (f_1, f_2, f_3)$, where $f_1 = \sqrt{3}(1,1,1), f_2 = \sqrt{6}(1,−2,1), f_3 = ...
1
vote
2answers
28 views

Orthonormal basis, decompse then add back

This is kind of a stupid question and I am taking some risk of getting some down-votes here, but, I can't resist posting it. Suppose $(u_1, u_2)$ is an orthonormal basis for $R^2$, and let $x$ be an ...
0
votes
1answer
50 views

Inner product space, orthonormal bases and change of basis.

I define unitary as $B*B=I$ I know that part (i) requires me to show the matrix coefficients are that of the inner product for bases A and B, however I am unsure how to get to this. Any help would ...
0
votes
2answers
33 views

Find the orthogonal complement of W

Consider the finite inner product space $\mathbb{R}^3$ where the inner product is the normal dot product. Find the orthogonal complement of W where W is $W = {(x,y,x+y) : x,y \in \mathbb{R}} \subset ...
2
votes
0answers
44 views

Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
0
votes
1answer
41 views

Question about maximal orthonormal subset in infinite dimensional vector space

The question is this: Let $A$ be an orthonormal subset of vectors in an infinite dimensional vector space $V$. Suppose for every $0\neq y \in V-A$ there is a $v \in A$ so that $\langle v,y ...
0
votes
1answer
53 views

Rows of orthogonal matrix from an orthonormal basis of $R^n$

The question is: Let $U$ be an $n \times n$ orthogonal matrix. Show that the rows of U form an orthonormal basis of $\mathbb R ^n $. So far I have stated: Since $U$ is orthogonal its column vectors ...
0
votes
1answer
21 views

Orthonormalising a Basis $B = (b_1 ,…,b_4 )$ so that it contains $u_1 = \frac{b_1}{\sqrt{\langle b_1,b_1\rangle }}$

I have just a small question: I have a scalar product and a Basis $B = (b_1, b_2, b_3, b_4)$. The task is to Orthonormalize the Basis B so that it contains $u_1 = \frac{b_1}{\sqrt{\langle ...
1
vote
3answers
82 views

Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
0
votes
1answer
26 views

Show that $T$ is a linear transformation given Orthonormal basis

Suppose that $T:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and suppose that $\{v_1,v_2,\cdots,v_n\}$ and $\{Tv_1,Tv_2,\cdots,Tv_n\}$ are orthonormal basis of $\mathbb{R}^n$. Prove that $T$ is a linear ...
0
votes
1answer
44 views

Linear Algebra Quick Question on orthonormal basis and inner product

I have a question asking to find an orthonormal basis of $p_2$ with respect to the inner product =2 X integral from 0 to 1 p(x)q(x)dx. What do I do with the 2 in front of the integral? When I solve ...
0
votes
1answer
58 views

Finding an orthonormal basis for the space $P_2$ with respect to a given inner product

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, ...
1
vote
1answer
44 views

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 ...
4
votes
2answers
42 views

Finding the dimension and basis of an orthogonal space

I am trying to find the basis and dimensions of the the space orthogonal $S$ which is in $\mathbb R^3$. $$S = \begin{bmatrix}1\\2\\3\end{bmatrix}$$ So the dimension would be two because it is $3 - ...
2
votes
3answers
118 views

Understanding the Gram-Schmidt process

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 ...
1
vote
0answers
39 views

Getting “semi” orthogonal basis from a linear independent set

Let $K_i: \mathbb{R}\mapsto \mathbb{R}^k$ are continuous functions for all $i=1,\dots,k-d$ such that for every fixed $t\in\mathbb{R}$ we have ${\cal K}_t=\{K_1(t),\dots,K_{k-d}(t)\}$ be a linear ...
1
vote
1answer
104 views

Eigenvalues of a $3\times3$ orthogonal matrix

Can anyone give me an example of 3x3 orthogonal matrix with complex eigenvalue.
1
vote
0answers
57 views

Will the projection of a singular matrix into an orthonormal space be non-singular?

I'm working through an implementation of the solution from 16.3.1 Dealing with the nullspace in the case of a singular within-class scatter matrix when performing discriminant analysis. In this ...
1
vote
1answer
55 views

Inner product question

We are given an inner product of $\mathbb R^3$: $f\left(\begin{pmatrix} x_1\\x_2\\x_3\end{pmatrix},\begin{pmatrix} y_1\\y_2\\y_3\end{pmatrix}\right) = ...
2
votes
4answers
78 views

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 ...
0
votes
1answer
28 views

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 ...
4
votes
2answers
44 views

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 ...
12
votes
3answers
459 views

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 ...
3
votes
3answers
81 views

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 ...
3
votes
0answers
63 views

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 ...
0
votes
1answer
77 views

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 ...
5
votes
2answers
96 views

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 ...
3
votes
0answers
88 views

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 ...
2
votes
3answers
108 views

$A$ 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 ...
-1
votes
1answer
55 views

Find conjugate transpose of linear transform

A difficult question I've been trying to tackle but I seem to hit a dead end. let $V$ be an inner product space over $\mathbb R$. We are required to find $T^{*}$ such that $<T(u),v> = ...
2
votes
1answer
75 views

SVD. why U and V have to be both orthonormal matrices?

I'm looking for the SVD factorization $A = U D V'$ starting from the set of equations $A u = v d$ and $A' v = u d$. Where u and v are vectors from the A and A' spaces and d the singular value. ...
1
vote
2answers
35 views

Finding a line of approximation using the normal equations for $A\vec{x}=\vec{b}$

To find the line $y=ax+b$ that best approximates the data points $\{(-2,3),(0,5),(1,7)\}$ I need to use the equation $$A\vec{x}=\vec{b}\ \ \ (\mbox{where}\ \vec{x}=\left({a\atop b}\right))$$ Then ...
0
votes
0answers
40 views

How do I normalize these vectors? Gram-Schmidt simple question

I was asked the following assignment: In the space $\mathbb R_3[X]$ with the inner product: $<f,g> = \int_{-1}^1 f(x)g(x)dx$, perform the gram-schmidt process to the basis: $1,x,x^2,x^3$. My ...
2
votes
1answer
405 views

Orthonormal Basis for Hilbert Spaces

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 ...
0
votes
2answers
28 views

How to normalize after Gram Schmidt process

I'm a bit confused about how to normalize after the Gram Schmidt process from my textbook. http://i33.photobucket.com/albums/d86/warnexus/length_zps0429d297.png I understand that to calculate the ...
0
votes
2answers
158 views

Find an orthonormal basis for the subspace of $\mathbb R^4$

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, ...
1
vote
4answers
92 views

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 ...
1
vote
1answer
29 views

Showing a set is orthonormal using an integral

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 ...
0
votes
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 ...
0
votes
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?
1
vote
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 ...
3
votes
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 ...
13
votes
4answers
369 views

Choosing an orthonormal basis in which a linear operator has a sparse matrix

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 ...
0
votes
1answer
51 views

finding projection on subspace

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} ...