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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2
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1answer
25 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 ...
1
vote
1answer
53 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?
0
votes
0answers
25 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 & ...
2
votes
2answers
18 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) ...
1
vote
0answers
21 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 ...
1
vote
1answer
33 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 ...
5
votes
1answer
81 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 ...
3
votes
1answer
41 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...
1
vote
2answers
30 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 ...
1
vote
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 ...
0
votes
1answer
13 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 ...
1
vote
0answers
38 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 ...
-4
votes
2answers
41 views

which is the inverse of this linear application? [closed]

$L:C[0,1]\rightarrow C[0,1]$ $L[f(x)]=\int_{0}^{x}f(s)ds$ 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 ?
1
vote
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 ...
2
votes
0answers
21 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 $U$ ...
3
votes
2answers
43 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 ...
0
votes
1answer
49 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 ...
2
votes
1answer
39 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\} ...
1
vote
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 ...
2
votes
1answer
24 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 ...
0
votes
0answers
23 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 $\theta_1$, ...
0
votes
1answer
65 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? ...
-1
votes
1answer
53 views
+50

Detecting linear dependencies in a matrix

Let $X$ be a matrix of $n$ rows (measurements) and $p$ columns (dimensions or features), and $n>p$. Denote by $r(i)$ the $i$th row of $X$. Assume that a subset of rows of $X$, denoted $r(i_j)$, ...
-2
votes
0answers
23 views

Why does $M$ have a limit rank in the operator norm? Why is $S$ bounded? [closed]

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 ...
0
votes
3answers
80 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?
0
votes
2answers
30 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 ...
0
votes
1answer
24 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 $A^{-1}$?
1
vote
1answer
25 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 $6det(2*A*(-7/2B)*B^{-1}A^{-1}) = 6det(-7)= 6*(-7)^6 = 705894$ ...
1
vote
2answers
38 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 ...
1
vote
2answers
51 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 ...
1
vote
3answers
39 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 ...
1
vote
1answer
27 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 ...
-1
votes
1answer
26 views

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 ...
0
votes
1answer
25 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
votes
3answers
74 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
votes
1answer
33 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 $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
7
votes
1answer
64 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
votes
1answer
31 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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votes
2answers
99 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 ...
1
vote
1answer
43 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 ...
4
votes
8answers
155 views

Is $y=mx+b$ linear?

Consider $f(x) = mx+b$. Let $b\ne 0$ If $f$ is linear, $f(0)$ should yield $0$ $f(0) = m(0)+b = b$ Therefore $f(x)=mx+b$ is nonlinear. Question: Why is $y=mx+b$ called a "linear equation"? ...
2
votes
3answers
87 views

Find the necessary and sufficient condition for $A^m\to0$

Let $A$ be $n\times n$ matrix on $\mathbb{C}$. Find a necessary and sufficient condition for $A^m\to0$ as $m\to\infty$. My thought: I think it should be that eigenvalues of $A$ are less than $1$. ...
-1
votes
1answer
23 views

Linear transformation, image [closed]

Linear space $\rm L$ is made from polynomials with real coefficients, whith maximum degree $2$. Is given basis $\rm e$ for space $\rm L: e_1 = 1, e_2 = x, e_3 = x^2$, also is given an image $\rm A$ in ...
0
votes
1answer
50 views

Decide the range of eigenvalues for $A+B$

Let $A,B$ be $n\times n$ Hermitian matrices on $\mathbb{C}$ such that all eigenvalues of $A$ lie in $[a,a']$ and all eigenvalues of $B$ lie in $[b,b']$. Show that all eigenvalues of $A+B$ lie in ...
0
votes
0answers
22 views

What is meant by “homogenous problem” exactly?

Let us look at an entirely linear problem with operator $L$. For an algebraic equation $Lu=0$ is a homogenous equation. If $L$ is a differential operator (PDE or ODE) it has to be supplemented with ...
1
vote
3answers
47 views

How come least square can have many solutions?

I know there always exists a least-square solution $\hat{x}$, regardless of the properties of the matrix $A$. However, I keep finding online that least-square can have infinitely many solutions, if ...
2
votes
2answers
50 views

Does a decrease of an entry of symmetric non-negative matrix decrease the norm?

I think the assertion would be false but I do not see an easy example. Or is it true obviously? Suppose we have a symmetric $A=[a_{ij}]$ with all entries non-negative. Now if we decrease a $a_{ij}$ ...
-1
votes
2answers
70 views

Proof that for all symmetric matrices $A$ and $B$, $AB=(BA)^T$.

Recall that a matrix, $M$, is said to be symmetric if and only if $M=M^T$. I've been trying to use the homomorphic nature of the transpose operator to prove this proposition but this approach hasn't ...
3
votes
3answers
47 views

Determine if a point is within a section of an octagon

I've been looking at other answers on this Exchange, such as this one. My math is fairly average, but working in Cartesian planes seems so long ago... My question is this: how to determine which ...
0
votes
2answers
17 views

Linear transformation, base change

Linear operator A in standard basis has matrix: 2 5 −3 −1 2 1 2 −3 2 Find its matrix in base f1 = (1, 1, 1), f2 = ...