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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Alternative more challenging problems to a simple supplementary/complementary problem

Original Question Find the measure of an angle if 80 degrees less than 3 times its supplement is 70 degrees more than 3 times its complement. The solution is simple. 3(180 - a) - 80 = 3(90 - a) + ...
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80 views

Proof of corollary of Farkas' lemma

I tried to prove the following lemma of Farkas' lemma: Given the system $Ax<b$, $A\in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^m$, the system is infeasible iff there exists $\lambda\in ...
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Does the theorem on transpose of a matrix extend to same dimension matrices?

In my textbook this theorem is stated on transpose of a matrix: Let $A$ and $B$ be $n \times m$ matrices, $C$ an $m \times k$ matrix, and $s$ a scalar. Then (a)$(A+B)^T = A ^T +B^T$ (b)$(sA)^T ...
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An error while solving a system of linear equations?

I know how to solve this with matrix operations. But I just seem to get them wrong. Please tell me where I'm wrong in my calculation: I don't need the full solution, only where I'm wrong.
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51 views

DFP rank-two update formula

when I am studying the DFP rank-two update formula, described as: $$B_{k+1}=(I-\rho_{k}y_{k}s_{k}^{T})B_{k}(I-\rho_{k}s_{k}y_{k}^{T})+\rho_{k}y_{k}y_{k}^{T},$$ where $$\rho_{k} = ...
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38 views

tight frame for $\mathbb{C}^N$

I have a question to ask Prove that if $K\in\mathbb{Z}-\{0\}$, then $\{\phi_p[n]=\exp(i2\pi pn/(KN))\}_{0\leq p<KN}$ is a tight frame of $\mathbb{C}^N$, i.e. $\sum_{k}|\langle f,\phi_p\rangle ...
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Operation count for Tridiagonal System

What is the operation count for solving the tridiagonal system $Ax=b$. I would guess it is $O(n^2)$ because all we are doing is making one sub-diagonal zero all the way across giving us $t(n)=n$ and ...
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Find the matrix for T with respect to the standard bases for different kinds

Define $D : R ^{2\times2} \rightarrow R^{2}$ by $D \begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} a \\ d \end{pmatrix}$ (a)Find the matrix representation of D with respect ...
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matrix of adjoint linear mapping

Let $U=\{u_1,...,u_n\} \subset E$ and $V=\{v_1,...,v_m\} \subset F$ be two orthonormal basis. If $a=[a_{ij}] \in M(m\;X \;n)$ is the matrix of the linear mapping $A:E \rightarrow F$, with respect to ...
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Jordan Normal Form and systems of linear equations

I am no expert in Linear Algebra, so the following question might be stupid or obvious. I am wondering if there is a connection between the solution space of a system of linear equation and its ...
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$\max_{1 \leq i \leq n}|\langle x,w_i\rangle|$, $\max_{1 \leq i \leq n}|\langle y,w_i\rangle|$ at same $w_i$ if $x$ and $y$ are close enough?

Let $x,y \in \mathbb{C}^n$ with $|x|_2 = |y|_2 = 1$. Let $w_1, \ldots, w_N \in \mathbb{C}^n$. Let $j,k \in \{1,\ldots, N\}$ such that $$ |\langle x,w_j\rangle| = \max_{1 \leq i \leq n}|\langle ...
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91 views

Symmetric Tridiagonal QR Algorithm

I have a question regarding QR algorithm. Suppose we are being given a symmetric tridiagonal matrix A (4X4) and perform QR factorization on A: A=QR. Then we define A':=RQ. A' still possesses the ...
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17 views

Is the following statement true (connection of basis and closed algebra)?

Let's have subspace A of square matrices in n-dimensional matrix space. They may have some properties (like traceless etc.). Imagine that they have $m$ independent components, $m < n^{2}$ (the ...
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17 views

Eigenbasis in infinite dimension

Here I computed eigenvectors of the shift operator. Assume the underlying field is $\mathbb R$. It follows that there are infinitely many eigenvectors (one per eigenvalue). Now I was wondering: Do ...
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72 views

Proving that integration and differentiation are linear maps.

Are the following approaches accurate for proving that integration and differentiation of polynomials are linear maps? Differentiation: Define $T \in ...
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38 views

Equations of motion for a block

I am looking for a very simplified derivation of the equations of motion (rotational and translational) for a block with a body fixed frame. I need to compare the EOMs for a system when the center ...
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98 views

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$ Which of the following is correct $?$ 1)Jordan form of ...
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30 views

Surjectivity of tuple of linear functionals

Let $V$ be a vector space (not necessarily finite dimensional) over $\mathbb{K}$ ($\mathbb{R}$ or $\mathbb{C}$), and $\phi_1$ and $\phi_2$ be two independent elements of $V^*$. Consider the map ...
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21 views

Problem about Kern and Image of a Linear Operator

Let $E$ be a vector space of finite dimension, and $A:E\rightarrow E$ a linear operator. Define a new operator $T_A:\mathcal L (E)\rightarrow \mathcal L(E)$, as $T_A(X)=AX$, $\forall X \in \mathcal ...
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playing with matrices

I have seen the proof of $M_n(R)$ (R - the field of Real Numbers) .has no two sided Ideals. The proof goes like the following. Let $I$ be an non zero ideal. We have to prove it is all of $M_n(R)$. ...
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28 views

A basic question on an unique vector identifying a linear functional in an inner product space

Let $f:V \to C$ be a non-zero linear functional where $dim(V)=l-1$. Write $V= kerf \oplus span\{v_0\}$ where $v_0 \perp ker f$. Now I am able to prove that there exists a multiple of $v_0$ (call it ...
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Solving systems of linear congruential equations

What is the best way to solve systems of the form $$ x_1 a_{1,1} + \cdots + x_n a_{1,n} = y_1 \mod m \\ \vdots \\ x_1 a_{N,1} + \cdots + x_n a_{N,n} = y_N \mod m $$ with $n \le N$, the parameters ...
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39 views

Geometric meaning of line equation in homogeneous coordinate

In Euclidean space, a line's equation is $$ax + by + c = 0.$$ While in homogeneous coordinates,it can be represented with $$\begin{pmatrix}x &y &1\end{pmatrix}\begin{pmatrix}a\\ b\\ ...
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26 views

Prove that $\mathbb{F^{\infty}}$ is infinite-dimensional

Prove that $\mathbb{F^{\infty}}$ is infinite dimensional. I'm not exactly sure how to do prove this. I was thinking of using something along the lines of "Suppose $\mathbb{F^{\infty}}$ is ...
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86 views

Find a common point that three lines meet.

I have a base 2D triangle with 3 lines coming out of each vertex with their own coordinate point (xyz) and a set distance, is it possible to calculate the specific point that they should meet? I also ...
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17 views

Add user in Center collection

I am working on a software which does the work basically counting points of user and generating scoreboard using some formula given below. It has got 2 paramaters x, y; ie, each user is assigned two ...
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26 views

Visualising $aw_1+bw_2=c \text{ where } a,b,c\in\mathbb R \quad w_1,w_2\in\mathbb C$

$aw_1+bw_2=c \text{ where } a,b,c,w_1,w_2\in\mathbb R, $ Is wel known and well studied by pupils very early on However I do not recall having seen $aw_1+bw_2=c \text{ where } a,b,c\in\mathbb R \quad ...
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The ith leading principal submatrix obtained by interchanging columns

Let $S \in GL_n(\mathbb{Q})$ be a non-singular symmetric $n\times n$ matrix with LDU decomposition. Let $L^T=U=(f_{ij})$ where $f_{ii}=1$ for all $i = 1, \ldots, n$ and $f_{ij} = 0$ if $ i > j$, ...
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self-adjoint subalgebras of matrix algebra

Is there any classification theorem for the self-adjoint matrix subalgebras of $M_n(\mathbb{C})$ the algebra of $n \times n$ matrices over $\mathbb{C}$ ?
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44 views

Maximum coordinate of a linear transformation of a vector

Given a vector $x \in R^n$ (variable) and a constant matrix $M \in \{0, 1\}^{m \times n}$ (known). $M$ is a binary matrix, meaning that its entries are either $0$ or $1$. I need to obtain an ...
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Correlating two time series or two random values independent from their order

I have a dumb question. Let's say I have two random variables X and Y with correlation a. I would like to re-correlate these guys with correlation b. As you know if I want to produce random ...
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66 views

Multi objective optimization into single objective.

I read that it is possible to convert a multi-objective optimization problem into single objective by using weighted sum method. I wanted to know if it is a good idea to convert a two objective ...
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25 views

If $B_A\cup B_B$ is a basis for $E$, then not necessarely $E=A\oplus B$.

Consider the vector space $\mathbb R^3$ and an endomorphism $f:\mathbb R^3\rightarrow \mathbb R^3$. Suppose we are given $A$ the matrix of $f$ relative to the canonical bases, and from which we ...
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Graph Theory and sandpiles

Using Matrix-tree theorem how could we conclude the order of S(G) is the sum of the weights of G's directed spanning trees into s where S(G) is the sandpile group of a sandpile graph G=(E,V,s).
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Finding the values of the coefficients from the following two equations.

We have these following two equations, where, and We need to calculate the terms ($A_1,A_2, B_1$ and $B_2$) so that for example, $p_1(R)$ and $A_1 p_1(q) + B_1 p_2(q)$ represent the same ...
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101 views

Solve system of matrix equations in finite field

I have the following system of matrix equations: \[ X_1 = A_1 X_2 B_1, \] \[ X_2 = A_2 X_1 B_2; \] where $A_i$, $B_i$ and $X_i$ are $n\times n$ matrices (for even $n$) over a finite field ...
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Finding out the range and statistical distribution

The range of the heights of the female students in a certain class is 13.2 inches, and the range of the heights of the male students in the class is 15.4 inches. Which of the following statements ...
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How to make a matrix less singular via linear transformation?

For a $m\times n$ matrix $$A = \begin{pmatrix}a_{11} &a_{12} &\cdots &a_{1n}\\ &&\cdots\\ a_{m1} &a_{m2} &\cdots &a_{mn}\end{pmatrix}$$, each column represents a ...
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66 views

On uniqueness of row reduced echelon matrices

Suppose R an R' are 2 x 3 row reduced echelon matrices and that the systems RX=0 and R'X=0 have exactly the same solutions. Prove that R = R' The way look at it is if we let A be the common solution ...
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28 views

Determine whether intersecting sphere moves towards cuboid?

I am programming a physics simulation in which I check every frame of a sphere intersects a cuboid. If it intersects, I want to check if the sphere moves "towards" the cuboid in a sense. If it does, ...
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29 views

The entries of a change of basis matrix

I am struggling to understand where the definition of the entries comes from in the following: Let $A=\{a_1, \cdots,a_m\}$ and $B = \{b_1,\cdots,b_n \}$ be ordered bases of a vector space $V$. Then ...
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Differences of skew symmetric matrices

Let $A$ be an invertible real skew-symmetric matrix, and consider the difference $A_R:=RAR^{-1}-A$, for orthogonal $R$. Is it true that $A_R$ is either zero or invertible? Does the answer depend on ...
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Monotone linear maps on symmetric matrices

I consider linear maps from real symmetric matrices in dimension n to real symmetric matrices in dimension m, monotone which respect to the ordering induced by the positive semidefinite cone (or, ...
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43 views

What does the hyperspace of 2d dimensions really look like?

I'm trying to follow a recipe on wikipedia (towards the end) to create some identically distributed samples from d statistical distributions: Generate an Nx2d sample matrix, i.e. each row is a ...
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Is there an analogue/primitive of PCA which can be in a metric space rather than a vector space?

Principle component analysis PCA is done in a vector space, basically projecting a given vector onto the eigen vectors of the covariance matrix. I'd like to think of a primitive analogoue of PCA, ...
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58 views

Invertible Composition of Maps

How can I show that if a composition of two linear maps is invertible, the two maps must also be invertible?
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Need to find components of a vector given distance and angle from a known vector on a known plane

I am looking for a way to compute the components of a vector (C in illustration) given: components of A and B, the angle between B and C, and the magnitude of C. I am looking for a way to solve this ...
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44 views

How is open set defined in linear map space

I got this statement in my homework: Prove that the invertible linear contractions are an open set in $Mat(2\times 2;\mathbb{R})$ I know what "invertible linear contraction" , "open set" and ...
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29 views

Gaussian Elimination $(Q-I)S = 0$, $Q$ is rotation matrix and $S$ is a vector

I need to solve this question where I have to compute $(Q-I)S = 0$ and do Gaussian elimination to solve for $S$. The problem is I do not know how to do Gaussian elimination on a rotation matrix. Theta ...
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20 views

How to set dihedral values to null?

I have a protein with many residues, but I would like to set the phi and psi angles of residue 15 to value of null. I have a file containing all residues and Cartesian coordinates, and I have another ...