All Questions
Tagged with vector-spaces orthogonality
268
questions
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]
$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
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3
votes
2
answers
114
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Nature of the Euclidean Norm
I've been re-reading my linear algebra book and a definition is given of the norm of a vector in $\mathbb{R}^n$ to be:
||v|| $= \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$.
For $\mathbb{R}^2$ and $\mathbb{...
- 197
1
vote
1
answer
74
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Orthogonal projection is bounded
Definition: Let $U$ be a subspace of $V$. The orthogonal projection of $V$ onto $U$ is the operator $P_U\in L(V)$ given by
$$P_U(u+w)=u$$ if $u\in U, w\in U^{\perp}$.
Let $V$ be a space with inner ...
- 1,494
2
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3
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104
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Why is $ R(A^*) \perp N(A)$ true?
Let a matrix the $A \in M_{n\times n}(\mathbb{C})$. My question is:
(1) Why every matrix $A$ satisfies
$ R(A^*) \perp N(A)$(where $R(A),N(A)$ are range of $A$,null space of $A$ respectively)?
And why ...
user1290851
1
vote
1
answer
91
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How orthogonal projection connects with eigen space?
I asked this question and asked to @JonathanZ how orthogonal projection relates with eigen space, he gives me following replies in comments:
Any time you have a subspace you can find an operator/...
user1290851
1
vote
1
answer
148
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Is union of orthonormal bases orthonormal?
Let a matrix the $A \in M_{n\times n}(\mathbb{R})$, and has set of eigenvalues, $\sigma(A)$={$\lambda_1$,$\lambda_2$........,$\lambda_k$}, that is
$\forall \lambda \in \sigma(A)$ such that orthonormal ...
user1290851
0
votes
1
answer
49
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Is the nullspace of transpose of any matrix orthogonal to the range of that matrix?
Let a matrix the $A \in M_{n\times n}(\mathbb{R})$. My question is why every matrix $A$ satisfies $R(A) \perp N(A^T)$(where $R(A),N(A^T)$ are range of $A$,null space of $A^T$ respectively)?
In ...
user1290851
0
votes
1
answer
59
views
Orthogonal orthornomal bases imply pair-orthogonal vectors
While self-studying linear algebra i started thinking about following problem: Let's say that $A, B \in \mathbb{C}_{n\times n}$ are orthogonal in a Frobenius sense orthonormal bases of complex vector ...
0
votes
1
answer
29
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Is $\space S^c\cup \{\vec{0}\}\space$ an accurate construction for the supplementary subspace of $S$?
Let $\space V\space$ be a vector space and $\space S\subseteq V\space$ a subspace. When considering the supplementary subspace i.e $\space W\space$ such that $\space V = S \oplus W$, is $\space W = \...
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1
answer
29
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Why there is no component when projecting a vector on an orthogonal space of another vector? [closed]
Based on the above diagram, may I know why there is no component in direction of vector v?
- 111
1
vote
1
answer
112
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Maximal totally isotropic subspace in a space with a degenerate skew-symmetric bilinear form
Let $V$ be a real finite-dimensional vector space with a skew-symmetric bilinear form $B \colon V \times V \to \mathbb{R}$. In general, we assume that the form $B$ is degenerate. A subspace $S \subset ...
1
vote
1
answer
58
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Proving the existence of specific scalars for perpendicular block vectors
$\newcommand{\ba}{\mathbf{a}}$$\newcommand{\bb}{\mathbf{b}}$$\newcommand{\bc}{\mathbf{c}}$
Given two vectors, $\mathbf{a}\in\mathbb{R}^{9}, \mathbf{b}\in\mathbb{R}^{6}$, each with a norm of $\lVert\...
- 121
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17
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Conceptual doubt with an orthogonal complement problem
I'm asked to find the orthogonal complement of $H\subset\mathbb R^4, H=\text{span\{(2,-1,0,0},(0,1,0,-2),(-8,0,0,8)\}$. But after doing the math, I got that the solution is a vector of the form $(a,2a,...
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1
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Is a vector $\mathbf{v}$ that lies in the null space of a rank-deficient matrix $A$ orthogonal to the rows and columns of $A$?
This might seem like a simple question, but I am quite confused by it.
I understand that a $D \times D$ rank-deficient matrix $A$ will collapse all vectors $\mathbf{x} \in \mathbb{R}^D$ to a flat ...
0
votes
1
answer
74
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Why Does the Existence of Eigenvalue 1 in Odd Dimensions does Extend to Even Dimensions? ($\mathbb{R}$ vector space)
Let $(V,\langle.,.\rangle)$ be a Euclidean vector space defined over $\mathbb{R}$ of odd dimension $n $ and let $f : V \rightarrow V$ an orthogonal mapping with $\operatorname{det}(f)=1 $. Then the ...
