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 solution of greatest possible dimension for this decoupled linear subspace problem?

Let $\left\{A_i\right\}$ be a $k$-element set of $n\times n$ Hermitian matrices, and let $P$ be an $n\times n$ rank-$m$ orthogonal projection matrix. We consider the projection of any matrix $A$ onto ...
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A question on the procedure of finding the matrix of a linear transformation of a polynomial and a combination of its derivatives

I'm trying to self-study Linear Algebra from Linear Algebra Done Wrong, but the book doesn't have solution manual so my question might be extremely easy, apologize in advance: The question is for the ...
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how would I represent supply/demand with augmented matrices?

I don't want to post the exact problem because then someone will just solve it and it's a practice problem but if I'm being too vague let me know. Theres 3 industries each with outputs expressed as a ...
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32 views

an inequality on matrix sign function : $A \leq \operatorname{sign} (A) \times A $

Let $A$ be an $n\times n$ real symmetric matrix, and $\operatorname{sign}(A)$ is the matrix sign function, which is defined as $\operatorname{sign}(A) := Z \pmatrix{-I_p & 0\cr 0 & ...
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170 views

Another proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
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40 views

Restricting solution $x$ to $Ax = b$ to natural numbers

Suppose that $A$ is $(n-1) \times n$ matrix that consists only of natural numbers (that is 0 and positive integers.) $b$ is $(n-1) \times 1$ matrix (vector) that only consists of natural numbers. For ...
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114 views

Root space decomposition

Regarding the direct sum of vector spaces/algebras, the dimensions of the parts of the sum should equal the whole. With the root decomp, the cartan sub algebra seems to have a dimension as high as the ...
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242 views

Calculate 3D-coordinates of a cube's points from the points on the projections

I have a following optical system: 3 cams (left and top, which is orthogonal to the left, and right, which is parallel to the left and orthogonal to the top) and the 2 cubes in the 3D-space with ...
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38 views

Definition of $F^\infty$

In his LA book Sheldon Axler defines the set of all sequences of elements of $F$ as: $$F^\infty = \{(x_1, x_2, \ldots): x_j \in F\text{ for } j = 1, 2, \ldots\}.$$ He also says: Sometimes we ...
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20 views

Why do we preform row reduction on $A^t$ for solving $A^tCAx = f$

So I'm trying to learn linear algebra using the MIT OCW lectures and while it can be difficult, I've been managing so far. I got to the 12th problem set and I'm stuck on the 2nd problem. I know one ...
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96 views

Axis aligned and non aligned ellipses and semi definite programming

Let's say I have a equation $$X_1^T \Omega X_1 =1 $$ $X_1$ is a $2\times 1$ matrix. $\Omega$ is a $2\times 2$ matrix. This defines an ellipse. $\Omega$ is a positive, semi definite, symmetric ...
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Minimum absolute determinant of a regular latin square matrix

It is easy to show that a latin square of size n x n has a determinant, which is a multiple of $\large \frac{n^2(n+1)}{2}$, if n is odd and $\large \frac{n^2(n+1)}{4}$, if n is even. This is a lower ...
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25 views

How to see that $\text{dim}(L)=k-1$?

Consider $L:=\left\{x\in\mathbb{R}^k: cx=\delta\right\}$ with $c=(c_1,\ldots,c_k)$ and $\delta\in\mathbb{R}$. Show that $\text{dim}(L)=k-1$. Do not know how to show that. Anyhow my first ...
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192 views

Proof that $S^\perp$ is a subspace of a vector space $V$

Just doing some review for a final exam and would like some feed back on the following proof if anyone would like to help me out. First the premise. Let $V$ be a finite dimensional inner product ...
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Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
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49 views

$tr(A)=0$ then exsists $P,Q$ such that $A=PQ-QP$ .

Let $\mathbb{F}$ be an arbitrary field and $A\in M_{n\times n}(\mathbb{F})$ such that $$tr(A)=0$$ Now show that there exists $P$,$Q$ $\in M_{n\times n}(\mathbb{F})$ such that $$A=PQ-QP$$ It is ...
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Solving the recursion $F(n)=K_0F(n-1)/(n-1)+K_1F(n-2)/(n-2)$

Please help me in solving the recursion $F(n)=K_0\frac{F(n-1)}{n-1}+K_1\frac{F(n-2)}{n-2}$, preferably using power series for the values of $F(n)$ in terms of $n$. Here $K_1$ and $K_2$ are ...
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47 views

How to find the rank of a toeplitz matrix?

Is there any trick to compute or estimate the rank of a toeplitz matrix ? Or is this still unknown for a general toeplitz matrix ?
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Intersection between $2$ lines (3D). This doesn't have a solution does it?

so I was looking through an old exam and this question was given: The teachers answer was the point $(9, -9, 21)$ I tried solving this myself, I got $x = x$, $y = y$, but I could not find a point ...
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64 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
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473 views

What is the difference between the span of a set to its subspace?

I am confused with some of the definitions of linear algebra. I know that the span of set S is basically the set of all the linear combinations of the vectors in S. The subspace of the set S is the ...
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45 views

Vector (scalar) product: associativity

Let $x$, $y$, $z$ be vectors of $\mathbb{R}^{n\times1}$. Consider this scalar result: $b = x^{\top} y z$. The issue is that the above product does not follow the classical associativity algerbra ...
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63 views

How to further simplify this equation?

Given that $V$ is an invertible $n\times n$ matrix and $\Sigma$ is a diagonal rectangular $m\times n$ matrix, $U$ is an $m\times m$ matrix, $b$ is an $m\times 1$ matrix and $\lambda$ is a positive ...
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104 views

Proof of Strong Duality via Farkas Lemma

I am trying to prove what is often titled the strong duality theorem. There is a hint in the book that I'm following, and I want to follow the method they have outlined for me. I will outline the ...
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50 views

Iterative methods for solving a linear equation system

There are several methods known for solving a linear equation system Ax = b (like Jacobi or Gauss-Seidel) by iterating $x_{n+1}=Mx_n+c$ with a matrix M, for which some norm is smaller than 1. But ...
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82 views

Integer matrices whose $m$-th power are identity matrix

How can one find all the matrices with integer entries of size $n \times n$ such that $A^{m}=I$ where $m$ is fixed integer and the matrix does not have fixed point in $\mathbb{Z}^n$ (except zero of ...
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42 views

Solving tridiagonal matrices where the top left element is zero

If I have a matrix like this: $$ \left[\begin{array}{rrrrrrrrr|r} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & ...
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31 views

change some element of a correlation matrix

I am working on correlation matrices. These matrices have the main property to be symetric , positive-semidefinite, have 1 on the diagonal and each of their elements is between -1 and 1. Let's say I ...
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75 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
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95 views

Eigenvalue bounds for a positive semidefinite matrix

I have a symmetric $(p\times p)$, positive semi definite matrix $\Omega$. If somebody says: find the eigenvalue bounds of the matrix such that $$w_1I \le \Omega \le w_2I$$ where $I$ is the identity ...
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39 views

Existence of Linear Maps and the Fundamental Theorem of Linear Maps.

Prove that there does not exist a linear map $T: \Bbb R^5 \to \Bbb R^5$ such that $\operatorname{range}(T) = \operatorname{null} (T)$. My proof goes like this: Suppose for the sake of contradiction ...
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find an ellipsoid given its intersection with axes and knowing the lengths of its principal axes

My question is about ellipsoids. I have an ellipsoid in 3D centered at zero so it has an equation: $x^T U \Sigma^2 U^T x = 1$ I know the lengths of it's principal axes (therefore the $\Sigma$ ...
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74 views

Can a ring of integers be free over a non-PID?

Let $K \subseteq L$ be an extension of number fields, and $A \subseteq B$ the corresponding rings of integers. $B$ is an $A$-module, generated by $[L : K]$ elements. If $K$ has class number one, ...
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33 views

How to import matrix from sif document

I want to make some computation (with scilab, scipy or other) over the matrix A of linear problems (in inequational form). Those problems are in .sif format (from the netlib library in fact) and I ...
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119 views

Generalized SVD, full-row rank and full-column rank.

In the generalized SVD (Golub and Van Loan, 2nd Ed. pp 471-473 Section 8.7.3), and its extension the Higher Order generalized ...
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39 views

having trouble with a 3-dimensional basis-change problem/

Let $V$ be a 3d vector space with a chosen basis $\alpha=\{e_1,e_2,e_3\}, \beta=\{f_1,f_2,f_3\}$ for $V$ satisfying: $$\begin{align}e_1 & =f_1+f_2+f_3 \\ e_2 &=f_2+2f_3 \\ e_3 & =f_3 ...
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49 views

Generator Matrix

I have a C in $F_2^6$ $(x_1,x_2,x_3,x_4) \to (x_1,x_2,x_3,x_4,x_1+x_2,x_3+x_4)$ for $x = (1,0,1,1)$ i get $c = (1,0,1,1,1,0)$ we know that $$c = G . x$$ G is the Generator Matrix in the solution ...
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44 views

Determining whether a function $\delta: M_{3\times 3}(\Bbb F)\rightarrow \Bbb F$ is $3$-linear

I've been working some suggested problems for a class, and I can't seem to understand the proper way how approach these problems. I tried to follow the following example and apply it to another ...
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57 views

Does this subgroup of $\mathrm{SL}(2,\mathbb{C})$ have a a name?

The set of matrices $g$ characterized by $g=\begin{pmatrix}a&ib\\ ic&d\end{pmatrix}$, where $a,b,c,d \in \mathbb{R}$ and $ad+bc=1$, can be easily shown to be a subgroup of ...
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Linear model for data that follow gaussian distribution

I have a question about linear regression. We have the linear regression of input data $(X,Y)=((x_1,y_1),(x_2,y_2)...(x_n,y_n))$ is $$F=aX+b$$ a,b are factors of the linear line, $y_i$ is {-1,1}. ...
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53 views

Linear w.r.t. any measure

Let $X$ be a Banach space endowed with a Borel $\sigma$-algebra. How do we call a real-valued Borel function $f$ that satisfies for any Borel probability measure $\mu$ the following formula $$ ...
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41 views

Identifying $\mathbb H^n$ with $\mathbb C^{2n}$

Let $X \in M_n(\mathbb H)$ (Hermetian field). It is possible to make $\mathbb H^n$ into a $2n$-dimensional vector space over $\mathbb C$, for example, by embedding $\mathbb C$ into $\mathbb H$ as ...
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necessary and sufficient conditions for a linear operator $T$ to be of a certain form

Let $V=\mathbb{C}$ as a real inner product space with the inner product defined by $(\alpha | \beta)=Re(\alpha\overline{\beta})$. For each $\gamma \in V$, let $M_{\gamma}$ be the linear operator on ...
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68 views

When is the LU decomposition unique?

I want to find out when a matrix decomposition $A = LU $ (L lower and U upper matrix) is unique? Clearly, if $A$ is not invertible, there is no chance that this decomposition is unique. Hence, ...
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62 views

Decide whether each map is an isomorphism( if it is an isomorphism then prove it)?

my problem is that i dont uderstand how they got the matrix for onto: f: M_2x2 --> P_3 can someone explains please?
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55 views

Polyhedron Basis

Given a Polyhedron formed by linear constraints $$(a^Tx <= b)$$ and you have given a orthogonal basis. Then every point inside the Polyhedron can be expressed by a linear combination of elements ...
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54 views

Hermitian solution(s) of Algebraic Riccati equation.

Given $3\times 3$ matrices $A,R,Q$ and $R,Q$ is skew-hermitian Find (all) hermitian solution(s) of: $$XRX+A^\dagger X-XA+Q=0$$. Generally I just need 1 (beautiful) solution.
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Getting coordinate vector in linear algebra

I know how to get the coordinate vector of single matrices by just joining them and doing a gauss jordan. But these are a 2x2, I don't know how to go about this, apparently no elimination can take ...
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75 views

every symmetric matrix can be orthogonality diagonalised

Let $A\in M_n (\mathbb R)$ be a symmetric matrix. It is known that $A$ is unitary diagonalizable. That means there exists some orthogonal matrix $P$ such that $$P^TAP=D$$where $D$ is a diagonal ...
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Numerical Computation for K smallest eigenvalues of a large Real Symmetric Matrix with restricted methods

I'm writing some code on a distributed platform, using some programming language like Hadoop, and now I need to calculate the K smallest eigenvalues for a Large Matrix. K is a small constant at most ...