Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Matrix Inversions

I have the following problem: "Suppose $P$ is invertible and $A=PBP^{-1}$. Solve for $B$ in terms of $A$." As far as I can tell, the value of $B$ depends of the values of both $A$ and $P$, not just ...
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Orthogonal Complement

"Let $\Bbb{V}$ be a vector space with an inner product $<\cdot,\cdot>$, and $S\subset\Bbb{V}$. We define the orthogonal complement of $S$, denoted by $S^{\perp}$, as follows: ...
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Determining Distance between two objects known in Size and distance of one object

I have a 10 Cent in my Hand (Diameter 19,2mm) and a DIN-A4 Paper on the table (297mm) I am holding the coin in front of my eye so that it fills the Paper, and i am using the following formula: x = ...
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isomorphic mapping on direct sum and products

I found an old post here: https://www.physicsforums.com/threads/question-about-isomorphic-mapping-on-direct-sums.709423/ While reading the answer to the question posted in the link above, I found ...
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Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
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Is my answer correct?

I'm trying to solve this question: My solution: Since $\varphi$ is continuous we have: $C\text{ is convex}\implies C\text{ is connected}\implies \varphi(C)\text{ is connected}\implies \varphi(C) ...
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Quaternions: Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship ...
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Show that B is singular

This is a linear algebra problem concerning singularity and linear independence. A is an $n \times n-1$ matrix where $A=\{A_1,A_2,...,A_{n-1}\}$ Show that $B$ is singular if ...
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Inverse rotation transformations

I'm taking the 2-degree gibmle system and position its alignment point in a arbitrary position (denoted by the axes angles phi for the first degree, and theta for the second). How can I reverse the ...
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Relation between Riccati Algebraic Equation and optimization problem

Reading this page: http://www.mathworks.com/help/robust/ug/minimizing-linear-objectives-under-lmi-constraints.html I got stuck in the result that says it can be show that minimizing Trace of X (a ...
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Let $Q$ be a symmetric $n$ by $n$ matrix, there exists an orthogonal matrix $F$ such that $F^TQF=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$

Let $Q$ be a symmetric $n$ by $n$ square matrix, there exists an orthogonal matrix $F$ such that $$F^TQF=\operatorname{diag}(\lambda_1,\ldots,\lambda_n),$$ with $\lambda_1,\ldots,\lambda_n$ being ...
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Find the dimension of a subspace by find a basis for the null space.

Below is the question and my proposed answer. It seems like it is a trick question, but maybe my answer is good enough or maybe I am wrong. Any help would be great. 2) Show that the dimension of the ...
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Does this formula take constant value?

Now, $x_i, \xi, f \in R^n(i= 1, 2, \cdots , k)$, and \begin{align} \sum_{i=1}^k x_ix_i^T\xi=f \end{align} holds. If the above equation is solvable about $\xi$, the value of $f^T\xi$ doesn't depend on ...
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Use of Matlab to put equation into vector form

Is there a way to put the following equation of a line into vector form using Matlab? $\displaystyle y=\frac{cos(s_n)-cos(s_{n+1})}{sin(s_{n+1}-sin(s_n)}(x-sin(s_n))-cos(s_n)$
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linear algebra question

Consider $n$ convex polytopes $S_1, \cdots, S_n$ and a set of matrices $W$ such that each matrix $A\in W$, we have that the $i$-th row of $A$ is a member of $S_i$. (In general $W$ is infinite.) ...
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Matrices over PID

Let $R$ be a PID and $A,B\in\operatorname{M}_n(R)$ are $n\times n$ matrices such that $\det(A)\sim\det(B)\neq0$,i.e., the ideals generated by $\det(A)$ and $\det(B)$ are the same, does there exist ...
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Finding the trace of $(I + \Sigma^{-1} AA^T)^{-1}$

I need to efficiently compute the trace of $$ B = (I + \Sigma^{-1} AA^T)^{-1} $$ where $\Sigma$ is diagonal and all its elements strictly greater than zero. $A$ is $-1$ on the diagonal and $1$ right ...
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slanting of non-square matrix

I'm looking for an operation to ''slant'' a (not necessarily square) matrix. I want this: $ \begin{bmatrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l ...
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Assigning a variable as constant and find out the minimum value

I have got a problem in assigning a variable as constant and solving a equations for minimum value. These are my equations: kd (2 h2 Cos[[Theta]2] - b2 Sin[[Theta]2]) (b1 - (b1 + b2) ...
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Why a matrix is a linear map

I know $L:\mathbb{R}^n \rightarrow \mathbb{R}^m $ by the formula $L(x)=Ax$ is a linear map. But I cannot understand why the matrix $A$, just itself, is a linear map. This question came to my mind by ...
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Fourier Interpolation

I have this Equation, that I modeled from my measurements and simulations. $I^{exp}_{l,m} = (\mathbf{F}^{H}.\mathbf{A}.I^{true})_{l,m}$; $H$ is the Hermitian transpose and $\mathbf{F}^{H}$ is a block ...
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Direct limits of free abelian groups and diagonalization

So, say I have a matrix $A\in M_d(\mathbb{Z})$ and would like to describe the group $\lim(\mathbb{Z}^d,A)$, i.e. the limit of the stationary system $$ \mathbb{Z}^d\to^A \mathbb{Z}^d \to^A ...
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Efficient algorithm to find a minimum spanning set for a given vector.

A few days ago a colleague proposed the following problem. Let $W$ be a finite subset of a vector space $V$, and let $v\in\langle W\rangle$ (the linear span of $W$). Is there an efficient ...
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Least squares fitting using cosine function?

Hello I am trying to fit a harmonic of the form $$y = b + c\cos(x)$$ to four data points (0,6.1) (.5,5.4) (1,3.9) (1.5,1.6) using least squares for homework. I know that the error $= Y_i - f(x_i)$ but ...
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Prove that if $S,T:E\longrightarrow F$ are linear transformations, then $\lvert r(T)-r(S)\lvert\le r(T+S)\le r(T)+r(S)$, where $r(T)=\dim Im(T)$.

I would like to know if this proof is right. In any case, anyone may feel free to provide a solution to the given problem. Prove that if $S,T:E\longrightarrow F$ are linear transformations, then ...
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Rational solutions to a system of equations

I have a system of equations $$\begin{align} xy + 3zw = 0; \\ xz + 2yw = 0; \\ xw + yz = 0. \\ \end{align}$$ Plugging it into a CAS, I see that all the rational solutions to this system have ...
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63 views

Differentiating a Quadratic Form

I'm having some trouble differentiating a quadratic form. I'm tasked with showing that $P(x) = \frac{1}{2} \left(b-Ax\right)^T C (b-Ax)$ is minimized by a vector $x$ satisfying $A^T C A x = A^T C b$. ...
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How prove this two condition is equivalent

Let $\alpha\neq \beta$ be nonzero column vectors in n-dimensional Euclidean space $\mathbb{R}^n$. Show that this follow two conditions are equivalent (1): $\alpha^T\beta>0$ (2): there exists ...
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Question about condition number $k$ of a matrix over a finite field

If $\lambda_{max}$, and $\lambda_{min}$ denote the maximum and minimum values of the eigenvalues of a normal square matrix repectively- are there any explicit bounds to the eigenvalues of such a ...
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Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$. I am able to write each event as a sum of distinct events that form a partition of the space. My goal is to find all the ...
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linear mixture word problem

Dan's Coffee Shop makes a blend that is a mixture of two types of coffee. Type A coffee costs Dan $\$5.85$ per pound, and type B coffee costs $\$4.30$ per pound. This month, Dan made $154$ pounds of ...
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linear combination of given basis vectors

Show that in an $n$-dimensional vector space $X$, the representation of any $x$ as a linear combination of a given basis $e_1,\ldots,e_n$ is unique. I only know that this can be proved by ...
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Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, $f(\textbf{a}+\textbf{v})=f(\textbf{a})+[Df(\textbf{a})]\vec{v}$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a function. Prove that if $f$ is linear, then for any $\textbf{a},\textbf{v} \in \mathbb{R}^2$, ...
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For the following system to be consistent, what must k not be equal to?

$6x - 4y + 4z = 5$ $9x - 6y + kz = -4$ $12x - 8y = -10$ Originally I just multiplied the first row by (3/2) and subtracted it from the second, which gives you a value of 6 for the answer. ...
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Is this matrix associated with an arbitrary group of events positive semi-definite?

Now I have an arbitrary group of events $X_1,X_2,\ldots,X_m$(with no independence or correlation assumptions, nor distribution knowledge), and define a symmetric matrix $\mathbf{K}$ as below: $$ ...
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Compute paths through graph vertex

I have a mesh, in which every vertex has most likely 6 to 8 neighbours. I need to compute like on the picture. For vertex O there are three incoming vertices: A, B and C. And we have these paths: AC, ...
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Calculating the left pseudoinverse of a Matrix whose columns are Probablity Mass Functions

I have a matrix $A_{m\times n}$, where $A_j$ , a column of $A$ represents a probability mass function, and so the sum over the column is 1. This is true for all the columns of A, i.e. $\forall j \in ...
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A question regarding the proof of Laplace's expansion on Wikipedia

I am reading the proof of Laplace's expansion on Wikipedia and have a dilemma regarding the following: $\tau = (n, n-1, \ldots, i) \;\sigma^\prime\; (j, j+1, \ldots, n)$ As far as I know, such ...
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70 views

Polynomial factοrisation

Problem Let $P(x)=x^n+64$ be a polynomial. Find the form of the natural number $n$ such that $P(x)=P_1(x)\times P_2(x)$, $\deg P_1(x),\deg P_2(x)\geq1$. I thought of taking $n \mod 4$. For ...
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Let $P_m$ be space of polynomials of degree $\leq m$ with real coefficients. Is it surjective?

Let $P_m$ be space of polynomials of degree $\leq m$ with real coefficients. Consider the application $$L:P_3 \to P_3, L(g)=(xg)^{'}$$ (i.e., L is the derivative of the product xg). Show that L is ...
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Cauchy-Schwarz problem (Maybe)

I believe that this may require Cauchy-Schwarz $$|\vec{u}\cdot\vec{v}|\leq |\vec{u}|\cdot|\vec{v}|$$ to solve. Let $y_1,y_2,\ldots ,y_p$ be $p$ positive numbers and let $i$ be a positive integer. ...
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Proof of uniqueness of reduced row echelon form

I've found a proof of uniqueness of reduced row echelon form. I have certian doubts with regard to this sentence: "It follows that R' and S' are (row) equivalent since deletion of columns does not ...
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Reconstruct a vector with a known vector and residual

I observe $\vec y \in \mathcal R^n$ and know $\vec x$. I assume that $\vec y$ mostly consists of $\vec x$, with some added residual $\vec r$. This gives me the problem $\vec y = a\vec x + \vec r$, ...
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Reducing a linear algebra expression to quadratic form

I am trying to solve the following exercise for my Machine Learning course. Expand this expression so that there are only quadratic terms: $(\mathbf{x} - \mathbf{\mu})^T \mathbf{\Sigma}^{-1} ...
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Preimage of an affine subspace

Let $V$ and $W$ be vector spaces. Prove: If $B$ is an affine subspace of $W$, and if $T \in \text{Hom}\,(V,W)$, then $T^{-1}[B]$ is either empty or an affine subspace of $V$. Would someone give me ...
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Could you help me with that Vector subspace proof 2

I need to prove that these S's are Linear subspaces of V. Could you help me with that Vector subspace proof? I tried: 1-If I choose any real number and to multiply by a negative value, can I say ...
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What is the relationship between three points on a quadratic curve and the curves coefficients?

In other words, is there a formula to get the coefficients a,b and c in terms of three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3, y_3)$? I am asking this because I have a linear algebra problem that ...
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$y=S y'$ implies $d^n y=|\text{det }S|d^n y'$?

An assertion is implicitly stated in a book without explanation: Given that: $S$ is an orthogonal matrix constructed by using the normalised eigenvectors of a symmetric matrix as its columns. $S$ ...
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Find $a_1, a_2$ and $F(x)$ for the given differential equation using the special solutions.

Suppose $y_1(x)= e^{-2x} + xe^{-x}$ , $y_2(x)= xe^{-2x} + xe^{-x}$ , $y_3(x)= e^{-2x} - xe^{-2x} + xe^{-x}$ are three special solutions to the differential equation, $y'' + a_1y' + a_2y = F(x)$ , ...
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Inverse of a matrix obtained by doubling every entry in column 1

Let B0 be the matrix obtained from B by doubling every entry in column 1 of B. Explain how could we obtain the inverse of B0 from the inverse of B. I know the answer of the inverse of B0 = the ...