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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Verification for a block-determinant evaluation, and some further thoughts

First, I want some verification for the validity of my approach for this det evaluation question: If $A,B\in M_n(K)$, $K$ is a number field (in the sense that $\Bbb Q$ is the smallest possible ...
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1answer
22 views

A program to visualize Linear Algebra?

I am asking here because I believe you have some idea of a good visualizer 3d program to see what are really: eigenvectors, subspaces, rowspaces, columnspaces and just answers on normal matrix ...
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12 views

Linear dependence of rank one approximations.

In my research, I am faced with the following problem. Let $\mathbf{X}$ and $\mathbf{Y}$ be two nonzero $m \times n$ matrices with entries in some field $\mathbb{K}$. Let $\phi:\mathbb{K}^{m\times ...
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0answers
8 views

Some “facts” on oriented angles in the Euclidean affine space of dimension 2

Let $\mathcal{E}^2$ be an Euclidean affine space of dimension $2$ oriented by $R=(O,B=\{u_1,u_2\}$. Let $$\mathcal{r}=\{P \in \mathcal{E}_2 \text{ such that } \overrightarrow{AP} \in \langle u ...
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1answer
12 views

Matrix operation: putting the rows next to each other

I have a matrix $A$ of dimension $N\times K$, and want to find a way to convert it to a matrix $B$ of dimension $1\times NK$. For example: ...
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0answers
14 views

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let's take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P ...
8
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2answers
94 views

Any neat way to calculate this Vandermonde-like determinant?

Let $x_i,i\in\{1,\cdots,n\}$ be real numbers, and $s_k=x_1^k+\cdots+x_n^k$, I'm asked to calculate $$ |S|:= \begin{vmatrix} s_0 & s_1 & s_2 & \cdots & s_{n-1}\\ s_1 ...
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37 views

Matrix of a differential equation

I had recentely encounter my first exercise about merging matrices notions and differential equations functions, but after solving the differential equation, I don't know how to represent it in the ...
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5 views

Computationally inexpensive method to find a rotation which minimizes the norm of two tensor difference.

So I have two matrices ${\bf T}_1$ and ${\bf T}_2$, they are tensors in the sense that they can be built as $${\bf T} = \sum_{\forall i} a_i({\bf v_i}{\bf v_i}^T)$$ with positive real weights $a_i$ ...
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28 views

Linear Algebra Subspace question, set difference

If $W_1$ and $W_2$ are subspaces of a vector space $V$, is $W_1 \setminus W_2$ ever a subspace of $V$? Why or why not? (Here $W_1 \setminus W_2$ denotes the set difference of $W_1$ and $W_2$: $W_1 ...
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19 views

How can we multiply partitioned matrices?

Im trying to perform the following product: Question: can we just think of each block as an element and perform the multiplication like matrix multiplication?
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46 views

Connecting a vector space to its dual - why?

Can someone explain to me - intuitively - why embedding a vector space into its dual should naturally fix its geometry? I mean, I can run the usual statements through my mind - "The injection into the ...
3
votes
3answers
51 views

Inner Product, Orthogonality, and Coordinate Systems

I remember my professor saying there are certain advantages to using an orthogonal basis. One is that it's easy to determine the coordinates of a given vector. For example, we are familiar with the ...
2
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2answers
37 views

How did they derive the image from kernel?

I understand its something to do with the rank nullity theorem, but im not sure how they applied it to get the basis of the image. By my understanding, they took the leading entries of the rows of ...
1
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1answer
23 views

Linear equation and linear differential equations

I remember noting from an algebra class that $x$ and $y$ of a linear equation neither divide or multiply with each other which is somewhat clear from the forms of linear equations: General form of ...
0
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1answer
30 views

What value to choose as representative of 100 votes?

I'm trying to use many votes to cast one vote. There are only 4 options, and I'm trying to use "consensus" to decide which to pick. 100 people vote. They can vote 1, 2, 3, or 4 stars. Distribution: 1 ...
3
votes
1answer
31 views

Making sense of $ f(y) - f(x) = \int_{\tau = 0}^{1} \langle \nabla f( x+ \tau (y - x)), y - x \rangle d \tau $

I was wondering if anyone has a good explanation why this holds. I came across this in the page 17 of this paper (equations at the end of the page): $$ f(y) - f(x) = \int_{\tau = 0}^{1} \langle ...
0
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1answer
36 views

triangular inequality

If we write $||a+b||\leq||a||+||b||$ explicitly in $\mathbb{R}^n$ it is $\sqrt{\sum^n_1(a_i+b_i)^2}\leq \sqrt{\sum^n_1(a_i)^2}+\sqrt{\sum^n_1(b_i)^2}$ how can it be if ...
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2answers
24 views

Finding the basis for a subspace given the span of a set of vectors

Let U = span{$u_1, u_2, u_3$}, where $u_1 = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} $, $u_2 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} $, $u_3 = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix} $, We ...
4
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1answer
56 views

“Hard” exercises on Linear Algebra and Analytic Geometry

I started lecturing this subject called "Linear Algebra and Analytic Geometry" and in the second day of class I was approached by an undergrad student, asking for referenced that would contain "hard" ...
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1answer
25 views

Understanding Composition Function (fg)(-1) for f(x)=x-3 & g(x)=x^2-8x+15?

Can someone help explain how to do the following composition function to me? (or at least get me started) Find the value of (fg)(-1) if ...
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votes
2answers
39 views

Using the harmonic mean to determine the time to perform a task with varying manpower. A/K/A “Frank, Bob and Diane” [on hold]

Each year West Coast Shipping provides transportation to Pebble Beach Concours D'elegance. Our 3 drives Frank, Bob and Diane can deliver all the vehicles from the auction to West Coast Shipping ...
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0answers
7 views

Good video course or Maple worksheets which help to understand linear algebra? [on hold]

I need a good video course or Maple worksheets which help to understand linear algebra at a university level?
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2answers
32 views

I'm having a problem with this number and digits problem. What to do?

During the last election , the total number of votes recorded in the municipality of San Juan was 8600. Had one-third of Estrada’s supporters stayed away from the polls and one-half of Arroyo’s ...
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2answers
33 views

Is the maximum of the eigenvalues of any symmetric positive?

Let $A$ be a symmetric matrix having dimension $n \times n \;, \; \; n\geq 2$. If one wants to pick the maximum of its eigenvalues, will the value be positive? Suppose A was an adjacency matrix, ...
0
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1answer
23 views

Evaluating a limit involving the power of specially structured matrix

Let $k\times k$ right-stochastic matrix $A$ be defined as follows: $$A=\left[\begin{array}{cccccc} p & 0 & 0 & \cdots & 0 & 1-p\\ 1 & 0 & 0 & \cdots & 0 & 0 \\ ...
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0answers
15 views

Finding a solution vector to linear system of equations with lowest hamming weight efficiently

I'm trying to solve a linear system of equations modulo 2. After performing gaussian elimination, I can get a solution of the form $v + c_1 \cdot n_1 + c_2 \cdot n_2 + \cdots + c_k \cdot n_k, c_i ...
0
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1answer
22 views

Is a symmetric diagonal matrix in which every entry is non-negative positive semidefinite?

Let $A$ be a symmetric diagonal matrix in which $(A)_{ii} \geq 0$. Should one conclude that this matrix is positive semidefinite?
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1answer
42 views

Linear Systems: Exponentials of a Matrix

I have a rather odd question to some, but one that has stumped me for a good few minutes on a homework assignment that states: For each matrix, find the eigenvalues of $\text{exp}{(A)}$, ...
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7 views

Properties of Linear Transformations

The book I am using is Differential Equations and Dynamical Systems by Lawrence Parko. Seeking to confirm my attempt at proving the following. Use the lemma in this section to show that if $T$ ...
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1answer
17 views

Linear Systems and Linear Transformation

I want to confirm my attempt to see if I am on the right track. The question is as follows. Show that the operator norm of a inear transformation $T$ on $\mathbb{R}^n$ satisfies ...
8
votes
1answer
73 views

What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like fourier transform - its actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
0
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1answer
29 views

How to solve the following system of equation?

Given two equations: $2mx+6y =1$ and $4x -(1-m)y = -16$ Find the value of $m$, such that the system has no solution? My attempt: From the first: $2mx = 1-6y$ Then $x= (1-6y)/(2m)$ Substitute ...
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2answers
43 views

Real Linear vs. Complex Linear

I recently started a new math course and got hung up on a particular problem from the book "Linear Algebra Done Wrong". Specifically, problem 1.3.6 (c). I am an engineer, and I believe I simply lack ...
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0answers
46 views

sign of eigenvalue [on hold]

Let L is a linear operator, we say it is stable if it has negative spectrum, why it is equivalent with there exists some $‎\varepsilon‎>0$ such that $$\langle Lh,h \rangle ...
1
vote
1answer
34 views

Confusion regarding dimension of a vector space

The dimension of a vector space is the number of elements in the basis for that vector space. If we look at $\mathbb R^n$, then we say that the dimension of $\mathbb R^n$ is $n$. So every element in ...
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0answers
8 views

Finding the Requested Value of Composition Functions?

I seem to still be having difficulty with understanding Composition Functions in Calculus and wondered if someone might be able to help explain in a way that will make the light bulb "turn on"? For ...
3
votes
2answers
41 views

Matrices such that $\det(p(A)-p(0))=p(\det A)-p(0)$ for all polynomial $p$

Question: Find all $2\times 2$ matrix $A$ such that $$\det(p(A)-p(0))=p(\det A)-p(0)$$ for all polynomial $p$. The zero matrix works since both sides are obviously zero. But I cannot find any ...
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votes
4answers
505 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
2
votes
1answer
11 views

Maximum of a Rayleigh quotient with non-Euclidean inner product

It's well known that, for a real and symmetric matrix $A$, $$ \max_v \frac { (Av,v) } {(v,v)} = \lambda_{\max}(A). $$ Now I'm looking at generalized Rayleigh quotients of the form $$ R = \max_v ...
0
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1answer
19 views

Simple exercise regarding space spanned by two vectors

Consider the vectors $X_1=(1,3,2)$ and $X_2=(-2,4,3)$ in $\mathbb R^3$. Show that the set spanned by $X_1$ and $X_2$ is given by $\{(Y_1,Y_2,Y_3):Y_1-7Y_2+10Y_3=0\}$.
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1answer
36 views

3 equations 4 unknowns.

I need to know what does this system describes please: $x+3t=2$ $y+t=-1$ $z+2t=1 $ Now subtracting equation (1) from (2) and (3) i.e (1)-(2)-(3) leads to $x-y-z=2$ hence the set of points of the ...
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0answers
38 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine: Basis We define the following basis: $$ A_n= ( \underbrace{00000...}_{n-1 times} 1 )^T $$ Hence, $$ A_1 =(111111 ... )^T $$ $$ A_2 = ...
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2answers
40 views

What is the matrix $\left[ DS(A) \right]$, which gives $\left[ DS(A) \right] H=AH+HA$?

In Hubbard's multivariable calculus book $DS(A):H \mapsto AH+HA$ is introduced as a linear transformation where $A$ is an $n \times n$ matrix, $S(A)=A^2$, and $D$ is the notation for derivative. It ...
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1answer
19 views

Linear Operators and Convergence

I am struggling with this question from my Differential Equations course: If $T$ is a linear transformation on $\mathbb{R}^n$ with $||T-I||<1$, prove that $T$ is invertible and that the ...
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1answer
20 views

Nonsingularity of submatrices

I'm still working on my question: Warm start of simplex algorithm after update of constraint matrix. While reading Schrijver's book "Theory of linear and integer programming" (reprint 1999) I ...
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1answer
12 views

Check if the equality holds

I have the following problem. For orthogonal $8\times 8$ matrix $M$ ($M\cdot M^{T} = 1$) check if the following equality holds $$ U = M^{T} \cdot \left( \begin{array}{cc} 1_{3\times 3} & 0\\ ...
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vote
1answer
18 views

Riesz representative of gradient of $f(u) = u^*u$ in different inner products

This is a seeming "paradox" that has been bothering me for some time, as it (or other situations like it) show up often when computing gradients for numerical optimization on complex vector spaces. ...
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3answers
25 views

Checking whether the result is positive definite or positive semi-definite with two methods

Given, $$A = \begin{bmatrix} 1 &1 & 1\\ 1&1 & 1\\ 1& 1& 1 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Using Method 1: ...
4
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0answers
21 views

Operator Norm of a Linear Transformation of a Matrix

The book I am using for the ODE course is Differential Equations and Dynamical Systems by Lawrence Perko. I am having a difficult time understanding what an operator norm of a linear transformation ...