Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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What is the difference between the scalar and vector components of a vector?

What is a scalar component of a vector and what is a vector component of a vector. suppose a vector is making and angle theta with the origin then in my book it is written that its x component is the ...
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15 views

Functions linearly independent and linearly independent gradients?

Let $F_1,...,F_n: \mathbb{R}^n \rightarrow \mathbb{R}$ be a set of $C^{1}$ functions. Is it true that they are linearly independent on a joint level set $\Omega:= \{ p \in \mathbb{R}^n; ...
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11 views

Relation between bilinear symplectic forms and symplectic matrices

1. Symplectic Forms Let $F : \mathbb{K}^{2n} \times \mathbb{K}^{2n} \to \mathbb{K}$ be a bilinear skew-symmetric nondegenerate form (as known as symplectic form). Then $F(u,v) = u^TAv$ where $A = ...
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1answer
10 views

Rotor: invariant under change of basis or not?

I wondered how the curl$$\text{rot}\mathbf{F}=\left( \begin{array}{ccc} \partial_y F_3-\partial_z F_2 \\ \partial_z F_1-\partial_x F_3 \\ \partial_x F_2-\partial_y F_1 \end{array} \right)$$of a vector ...
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1answer
22 views

Why the Householder matrix is orthogonal?

A Householder matrix $H = I - c u u^T$, where $c$ is a constant and $u$ is a unit vector, always comes out orthogonal and full rank. Why $H$ is orthogonal (looking for an intuitive proof rather than ...
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17 views

Question about the coordinates in a new origin on the plane.

I'm reading a book on analytic geometry, specifically on a chapter on change of coordinates. It says that having the origin $O$, one point $P$ and a new origin $O'$, the vector that describes the ...
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3answers
28 views

Show that $\ker \hat{T} = \text{ann}(\text{range } T)$

This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by ...
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48 views

Representing a vector in $\mathbb{R}^{3}$ as sum of only two vectors in $\mathbb{R}^{3}$

Is it possible? Or more generally can any vector in $\mathbb{R}^{n}$ can be represented as sum of (n-1) or less vectors in $\mathbb{R}^{n}$? -----EDIT----- What I basically want to ask is that can ...
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2answers
25 views

A theorem of symmetric positive definite matrix.

Is the following true? Let $g=(g_{ij})\in M(n,\Bbb R)$ be a symmetric positive-definite matrix and let $a=(a_1,\ldots,a_n)\in\Bbb R^n$ be any vector. Then, $$v^Tgv=1\implies (v\cdot a)^2\leq ...
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1answer
23 views

Is convex hull linear subspace of linear hull?

We have some convex and compact supset $G$ of banach space $B$ and finitely many points ${x_1,...,x_N}$ . The question is : does the convex hull $C$ of ${x_1,...,x_N}$ a linear subspace of space ...
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1answer
38 views

Does $A$ and $(A+I)^{-1}$ commute for positive operator $A$?

Suppose that $A$ is a bounded positive operator ($A \geqslant 0$) on some Hilbert space. Can I say that $A$ and $(A+I)^{-1}$ commute?
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21 views

Matrix representation of another matrix

Let $\mathbf{c}\in \mathbb{R}^n$ and $\mathbf{X}(s)= \begin{bmatrix} X_{11}(s) & X_{12}(s) & \cdots & X_{1n}(s) \\ X_{21}(s) & X_{22}(s) & \cdots & X_{2n}(s) \\ \vdots & ...
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16 views

Compressive sensing for complex matrix

I'm fairly new to compressive sensing, and I have been looking for a MATLAB implementation of the problem $$ A x = b $$ where $A$ is non square, $x$ is kind of sparse and all the numbers involved are ...
2
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2answers
15 views

For certain positive semidefinite matrices, subtracting the outer product of their row-sums does not change the positive semidefiniteness

Let $e$ denote the vector of all ones, $J=ee^T$ and $\langle A,B\rangle = trace(AB^T)$. Consider a symmetric positive semidefinite (psd) matrix $A\geq 0$ (that is, $a_{ij} \geq 0$ for all entries) ...
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0answers
13 views

Camera calibration: how does checkerboard size/numbers/placement affect accuracy

I am trying to calibrate a camera using a checkerboard by the well known Zhang's method followed by bundle adjustment, which is available in both Matlab and OpenCV. There are a lot of empirical ...
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1answer
27 views

Underdetermined vs Overdetermined Problem

I'm trying to create a model which is of the form $$y = (a_0 + a_1l)[b_0+\sum_{m=1}^M b_m\cos(mx-\alpha_m)] [c_0 +\sum_{n=1}^N c_n\cos(nz-\beta_n)]$$ In the above system, $l$,$x$ and $z$ are ...
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1answer
56 views

Determinant of the Transpose of an Operator.

Let $V$ be a vector space over a field $F$ of characteristic $0$. A linear operator $T$ on $V$ induces a linear operator $\Lambda^k T:\Lambda^k V\to \Lambda^k V$ such that $\Lambda^k T(v_1\wedge ...
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22 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of several simplicies, with all coordinates being non-negative. That is, given $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to ...
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2answers
27 views

How to solve the system of equations $\{10^{-4}x_1+x_2=1, x_1+x_2=2\}$ using finite precision arithmetic with three significant figures?

Consider the following two equations: $10^{-4}x_1+x_2=1$ $x_1+x_2=2$ Solve using Gaussian Elimination using finite precision arithmetic with three significant figures. I'm a little ...
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2answers
56 views

If $AB$ and $BA$ are defined, then $AB$ and $BA$ are square matrices.

So as self-practice, I'm going over some proofs from Linear Algebra. I came across the following proof: $$\text{Prove that if both products $AB$ and $BA$ are defined, then $AB,BA \in M_{n,n}$.}$$ I ...
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1answer
12 views

Determine Orthogonal and non orthogonal using Coordinates

Can we identify using coordinates that if Polygon is orthogonal or non orthogonal. data = [(100, 100), (100, 200), (300, 200), (600, 400), (1150, 400), (1150,300), (600,300), (300,100)](These ...
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35 views

show the following equivalence

Let $A = \left(a_{ij}\right) \in C^{n \times n}$ be a self-adjoint matrix (that is, a matrix such that $A^\ast = A$. Show that $A$ is positive definite if and only if the determinant of the matrix ...
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40 views

which is the inverse of this linear application? [on hold]

$L:C[0,1]\rightarrow C[0,1]$ $L[f(x)]=\int_{0}^{x}f(s)ds$ that is linear and R(T)=$C^{1}[0,1]$ s.t $L(0)=0$. I need calculate $L^{-1}: R(T)\rightarrow C[0,1] $ I could give some suggestion ?
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2answers
28 views

Grasping “Substitution” in terms of linear algebra

So I have a set of equations: $$x_{1} + x_{2} = 1$$ $$x_{2} + x_{4} = 3$$ From linear algebra, we know that (say, we're in $\mathbb{R}^{4}$, i.e. we have 4 variables), the solution space to the ...
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0answers
19 views

Determinant 1 matrix does not change p-adic measures

Let $f:\mathbb Z^d \rightarrow \mathbb Z^d$ be a linear map having determinant 1. Is there an obvious way to see that if $U\subseteq \mathbb Z_p^d$ is a measurable set, then the p-adic measure of $U$ ...
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2answers
41 views

Show $f=f^*$ including inner product

Let $V$ a $\mathbb{C}$-vector space with inner product $\langle \cdot , \cdot \rangle$ and $f:V\to V$. Show that if $\langle f(v),v\rangle\in \mathbb{R}$ for $v \in V$, then $f=f^∗$. I was thinking ...
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1answer
35 views

Matrix with entries equal to $1$ and $-1$ (Sign Matrix)

What can we say about the determinant and (or) maximum eigenvalue of a matrix with entries equal to $1$ and $-1$. Further assume that the rows and columns are linearly independent. Are there special ...
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1answer
38 views

Finding a basis for $V, W, V+W$ and $V \cap W$

Problem: Let \begin{align*} V = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid y+z+u = 0 \right\} \end{align*} and \begin{align*} W = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid x+y = 0, z = 2u \right\} ...
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1answer
11 views

Use of GS before projecting a vector onto a plane

I need help with the following exercise: Given the vectors $u_1 = (2,-1,2), u_2 = (1,2,1), u_3 = (-2,3,3)$, what is the projection of $u_3$ onto the plane spanned by $u_1$ and $u_2$. I'm not sure if ...
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1answer
20 views

Find the elementary divisors of a matrix given its characteristic and minimal polynomials

This question comes from and old exam: Suppose the square rational matrix $A$ has characteristic and minimum polynomials $p_A(x) = x^6(x^2-2)^3(x^2+4)^2$ and $m_A(x) = x^2(x^2-2)(x^2+4)^2$ and $null A ...
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22 views

Span of a projection matrix $P(\theta, \phi)$

I have a projection matrix which depends on two parameters, $\theta$ and $\phi$. I am interested in finding out if the relationship between space spanned by the projection matrix for say $\theta_1$, ...
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1answer
59 views

Characteristic polynomial of A, if $\det(\operatorname{adj}(\operatorname{adj}(A))) = 81$?

Let $A$ be a square real matrix whose eigenvalues are positive integers, with $$\det(\operatorname{adj}(\operatorname{adj}(A))) = 81 \, .$$ What is the characteristic polynomial of A? Any hints? ...
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1answer
29 views

Meaning of points in 3D space that lie on a line

$\{X_i\}_{i=1}^n$ are $n$ points in a three dimensional space with basis functions $\{\phi_i\}_{i=1}^3$. For simplicity we can assume that this the space is $\mathbb{R}^3$ with the natural basis ...
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Why does $M$ have a limit rank in the operator norm? Why is $S$ bounded? [on hold]

Define operator $S$ and $M$ on $\ell^2$ by $(SX)_n = \begin{cases} 0 & n = 0 \\ x_{n - 1} & n > 1 \end{cases}$ $(Mx)_n = \dfrac{1}{n + 1} x_n,\qquad n \ge 0$ Why does $M$ have a limit ...
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78 views

Are there nontrivial vector spaces with finitely many elements?

I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space?
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2answers
28 views

Canonical linear mapping is bijective

Let $V$ be a $K$-vector space with finite dimension. Proof that mapping: $V^* \otimes V \rightarrow {\rm End}_K(V), \ h\otimes a\mapsto (x\mapsto h(x)a)$ is bijective. So we have one mapping, which is ...
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1answer
23 views

Is there an explicit formula for $\left(xx^T\right)^{-1}$ with $x\in\mathbb{R}^n\setminus\left\{0\right\}$?

Let $x\in\mathbb{R}^n\setminus\left\{0\right\}$. Obviously, $$A:=xx^T$$ is symmetric and positive definite. Hence, $A$ is invertible. Can we find an explicit formula for $A^{-1}$?
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1answer
24 views

Calculating Determinant Using an Equation

$detA_{6x6} \neq 0$. $2A+7B=0$ Calculate $6det(2(A^t)^2B^{-1}A^{-1})$ My solution attempt: $A = -7/2*B$ and $det A^t = det A$ so $6det(2*A*(-7/2B)*B^{-1}A^{-1}) = 6det(-7)= 6*(-7)^6 = 705894$ ...
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2answers
34 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
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2answers
49 views

Linear algebra: What is the difference between homogenous and particular solutions?

First, I would like to mention I'm new to asking questions here, though I have found many answers here! I hope to get more involved here over time, I really like this site. If you have any suggestions ...
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3answers
32 views

How to compute the projection of a polyhedron

Suppose that we have a polyhedron in $(x,y)$: $P=\{ (x,y) \mid A_1 x +A_2 y \leq b \}$ How can I find the polyhedron $P_x=\{ x \mid (x,y)\in P \}$? In other words, I would like to write $P_x=\{x ...
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1answer
24 views

How do I extrac the anisotropic part of a tensor?

Given the elements $\chi_{ij}$ of a tensor in cartesian coordinates, with \begin{pmatrix} \chi_\bot& 0 &0 \\ 0 & \chi_\| &0\\ 0&0 & \chi_\| \end{pmatrix}, where the ...
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1answer
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Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
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1answer
24 views

Tensor product of 2 vectors

Let be V,W 2 K-vector spaces and $a \in V, b \in W$.a,b are vectors. We know that $a\otimes b=0 \in V \otimes W$. Proof that a=0 or b=0. From definition it will be a matrix with elements $a_i b_j$ and ...
5
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3answers
58 views

Optimal approximation of quadratic form

Let $\mathbf{x}\in\Bbb{R}^n$ and $A\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes the space of symmetric positive definite $n\times n$ real matrices. Also, let $Q\colon\Bbb{R}^n\to\Bbb{R}_{+}$ be ...
2
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1answer
31 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
6
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1answer
59 views

Group action on a category

Motivating example: We get a functor from the category of real vector spaces to the category of complex vector spaces by complexifying (i.e. tensoring over $\mathbb{R}$ with $\mathbb{C}$). Let ...
2
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1answer
28 views

The deconposition of $\mathfrak{so}(V \oplus V^*)$

Let $V$ be an n dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ in $V\oplus V^*$ such that $(v+\xi , ...
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2answers
95 views

What is that matrix?

Let an inner product on $\mathbb{R}^n$ be given by its Gramian matrix $G$. Let $A:\mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear operator with $\mathop{\rm rank} A=k$ (We denote its matrix also by ...
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1answer
40 views

How to express double orthogonal complement?

Let $V$ be a Hilbert space and $U \subseteq V$. Then $U^\perp = \{\mathbf{v} \in V|\forall \mathbf{u} \in U, \langle \mathbf{u}, \mathbf{v} \rangle = 0 \}$. My question is, how do you express ...