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 connected components of pseudospectra of matrix polynomial? .

What is connected components of pseudospectra of matrix polynomial? thanks.
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Sketching phase portrait of an ellipse

I have a system of linear ODE's as follows: $$\frac{dx}{dt} = y, \frac{dy}{dt} = -4x$$ which has solution $$\begin{bmatrix}x\\y\end{bmatrix} = \alpha\begin{bmatrix}cos2t\\-2sin2t\end{bmatrix} + ...
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1answer
11 views

Assigning a specific value to components of a vector

So far, I've run into this twice and I'm not exactly sure how to make this connection myself, but in this case, I've been asked to find the dot product of $(i+j+k) \cdot (3i+2j-5k)$ I understand ...
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0answers
20 views

How to prove this decomposition

There are two vectors l=$(l_1,l_2)^T$, m=$(m_1,m_2)^T$, and a symmetric matrix S=$\begin{bmatrix}s_{11}&s_{12}\\s_{12}&s_{22}\end{bmatrix}$. Then, ...
2
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2answers
57 views

Proving a Set is a Vector Space

for each $\left(x_1,y_1\right)\,,\, \left(x_2,y_2\right)$ that is an element of $\mathbb R$x$\mathbb R$ define $$(x_1, y_1) + (x_2, y_2) = (x_1 +x_2 + 2 , y_1 +y_2 -3)$$ And, for each $(x,y)$ that is ...
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1answer
29 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
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1answer
21 views

Using a linear function as a routine to determine a matrix

Let $F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ be a linear function, i.e., $$F(\alpha x + \beta y) = \alpha F(x) + \beta F(y)$$ Suppose you are given a routine that returns $F(x)$ given any ...
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3answers
43 views

A jazz concert brought in 128,000 on the sale of 8,100 tickets. If the tickets sold for $10 and $20 each, how many of each type ticket were sold?

I am currently struggling on how to figure this out. I got as far as 165,000-81,000=84000. I am unsure what to do next. Thank you in advance!
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0answers
11 views

Strange property of correlation matrices

Let $\mathbf{C}$ be a statistical correlation matrix. This implies that $\mathbf{C}$ is a Hermitian, non-negative definite, $n\times n$ matrix with ones along the main diagonal and with off-diagonal ...
4
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1answer
18 views

Diameter of unitary group.

Define a function$$N: \text{End}_\mathbb{C} \to \mathbb{R}_{\ge 0},\text{ }N(a) := \max_{\{v \in V\,:\, |v| = 1\}} |av|.$$ What is $$\max_{a, b \in U(V)} N(a - b),$$the "diameter" of the group ...
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14 views

Constrained optimization with several equality constraints

In maximizing a function of $n$ variables with $m$ equality constraints, it is required that the Jacobian derivative of constraints has full rank at optimal points. Can some one provide me with the ...
0
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1answer
27 views

Linear endomorphisms of $k(t)$

Let $k$ be a field and let $k(t)$ denote the field of rational variables in $t$. Is it possible to characterize all $k$-linear transformations from $k(t)$ to $k(t)$? Is $End_{k}(k(t)) \cong k(t)$ ?
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3answers
28 views

matrix with all rows positive

I am thinking about a problem in a different area than linear algebra, but I came across a matrix with sum of entries of all rows positive, i.e. a matrix $A$ such that $\sum_{j} A_{ij}>0$ for all ...
4
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1answer
19 views

Why do polynomial regressions have larger variance at the end?

In reading the book "An Introduction to Statistical Learning with Applications in R", I came across this graph: It shows that the point-wise variance is larger at the ends of the regression curve. ...
2
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2answers
30 views

About two systems of linear equations defining the same linear subspace

Assume that we have in a linear space $X$ a linear subspace $V$ defined system of linear equations : $$ f_1=0, \\ f_2=0, \\ \,\vdots \\ f_k=0, $$ where linear functionals $f_1,\ldots,f_k \in X^*$ and ...
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0answers
25 views

Specific Root of Interpolating Polynomial

We define polynomial $P=(x-\alpha)\cdot g(x)$, where $deg(p)=n-1$, $\alpha \leftarrow \mathbb{Z}_p$. We evaluate $P$ at some $x_i$ values. So we get $(x_1, y_1),...,(x_n, y_n)$, where $P(x_i)=y_i$. ...
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2answers
51 views

Can Two Different Polynomials Agree on an open interval? [duplicate]

Question: For a high degree polynomial $P_1$ , can we have another polynomial $P_2$ that is a part of $P_1$ (or they agree on open interval)? TBN: This question is partially answered in ...
2
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1answer
25 views

$U(\mathbb{C}^n)$, $SU(\mathbb{C}^n)$ connected subsets of $M_n(\mathbb{C})$?

As the title suggests, is $U(\mathbb{C}^n)$ a connected subset of $M_n(\mathbb{C})$? How about $SU(\mathbb{C}^n)$?
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1answer
19 views

Do approximate identities remain approximate identities if one adjoins 1 to a C* Algebra?

If we have a C* Algebra $\mathscr{U}$ without an identity we can adjoin an identity $\mathbb{1}$ in the following way: We take $\mathscr{\tilde U}$ to be the set $\{(\alpha,A); \alpha \in ...
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2answers
25 views

Let $V=\mathbb{R}^\mathbb{R}$, let $W$ be the subset of $V$ consisting of all monotonically inc or dec functions. Is $W$ subspace of $V$?

Let $V=\mathbb{R}^\mathbb{R}$ and let $W$ be the subset of $V$ consisting of all monotonically-increasing or monotonically-decreasing functions. Is $W$ a subspace of $V$? Any solutions or hints are ...
2
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1answer
31 views

Does multiplication by a positive definite matrix preserve eigenvalues?

Let $A$ be a positive definite matrix and let $B$ a matrix. Then, $AB$ is similar to $A^{\frac{1}{2}}BA^{-\frac{1}{2}}$, which is in turn similar to $B$, so I get that $AB$ and $B$ are similar. ...
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3answers
67 views

Is it right to say that if two vectors, $A$ and $B$, have same $L^p$ norms, for all $p$, then $A = B$?

Is it right to say that if two vectors, $A$ and $B$ (all elements of $A$ and $B$ are positive), have same $L^p$ norms, for all p, then $A = B$ ?. Thanks.
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0answers
25 views

Recovering a basis from an isomorphism with the dual space.

Let $V$ be a finite dimensional vector space, then given a basis for $V$ constructing an isomorphism $V \rightarrow V^*$ is easy, but how about the reverse direction? Given an explicit isomorphism ...
2
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4answers
51 views

Vector Functions of One Variable

Question A particle moves along the curve of the intersection of the cylinders $y=-x^2$ and $z=x^2$ in the direction in which $x$ increases. (All distances are in cm.) At the instant when the ...
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2answers
37 views

Rank of the product of two full rank matrices

I have searched for the above topic and found some results, but the answer I am looking for is not found anywhere. Here is my question: Given $A_{m \times n}$ matrix with rank $m$, and $B_{n ...
2
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0answers
14 views

Does there exist nonzero eigenvector such that $E(v) = 0$?

This is a followup to my previous question here. Let $\text{U}_+$ be an associative $\mathbb{C}$-algebra with two generators $E$, $H$, and one defining relation $HE - EH = 2E$. Let $M$ be a ...
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1answer
57 views

Overlapping Polynomials

This question is related to this:Interpolating Polynomial & It's Root We have $P_3=P_2\cdot P_1$,for three non-zero polynomials. The degree of each polynomial is at least 1. Question: Does ...
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12 views

Warm start of simplex algorithm after update of constraint matrix

Assume we found an optimal solution $\mathbf{x}_1$ of the linear program \begin{gather} \max \mathbf{n}^T\mathbf{x}\mbox{ s.t. }A\mathbf{x} \leq \mathbf{b}\tag{1} \end{gather} using the simplex ...
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0answers
6 views

About the algebra used in linear discriminant analysis in scikit learn (LDA using SVD)

I've looked for info about how LDA is impemented in scikit-learn but there's no clue about what I'm looking for. In this code in python: ...
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0answers
9 views

Probability That a Polynomial has Specific Root when we use Permutation Polynomial

To some extent similar question was asked here: Polynomial Interpolation and Security Imagine we have $\vec{x}=(x_1,...x_n)$ and two polynomials $P_1$ and $P_2$. Degree of $P_1$ is fixed $n-2$, ...
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17 views

Write down the 5 equations Cx = b. Find a combination of left sides that gives zero(5x5 matrix)? [on hold]

The very last words say that the 5 by 5 centered difference matrix is not invertible. Write down the 5 equations Cx = b. Find a combination of left sides that gives zero. What combination of b1, b2, ...
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1answer
32 views

Eigen values of A*A are non negatives.

If $A$ is a complex matrix of order $n$ then i like to prove that all eigen values of $A*A$ are non negative where $*$ is transpose conjugate . $ \lambda \|x\|^2 = \langle \lambda x,x \rangle = ...
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1answer
33 views

Boundedness of matrix norm

Let $A$ be a n by n matrix whose entries are continuous functions of $x\in \mathbb{R}^n$. Fix a matrix norm $\|\cdot \|$ and assume that $\|A(x^\star)\| < 1$. Then, the claim is that there exists ...
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2answers
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$W$ a subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$?

Let $W$ be the subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$? I'm not sure how to do this. Any ...
2
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1answer
22 views

Finding an equilibrium solution to a first order system of equations.

Given a model: $ y''+\alpha y'+\beta y + \gamma y = -g $ I can see that it can be converted to a system of first order equations as follows: $y_{1}=y$, $y_{2}=y'$ and as such $y_{1}'=y'$ and ...
1
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1answer
38 views

Let $F$ be the set of all functions of the form $f$: $t\to \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$. Is $F$ an integral domain? Is it a field?

Let $F$ be the set of all functions $f$ from $\mathbb{R}$ to itself of the form $f$: $t\to \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$ where $a_k$ and $b_k$ are real numbers and n is a natural number. ...
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0answers
5 views

formula to calculate bounding coordinates of an arc in space

I have an arc in space with known 2 endpoints x1,y1 and x2,y2 centrepoint x3,y3 radius r What would be the formula to find the coordinates of a box that fits the limits of the arc.
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4answers
56 views

Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.

I try to transform Transform $$f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$$ to a diagonal form. I can do it using eigenvalue, but when I directly complete the square to find its ...
2
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2answers
38 views

The norm of a bounded linear operator has this formula: $\|T\| = \sup_{\|v\| = 1} \|T v\|$

Trying to prove $\|T\| = \sup_{\|v\| = 1} \|T v\|$, given $\|T\| := \inf_{C \geq 0} \{C: \|Tv\| \leq C\|v\|\}$. I know that $\|T(v)\| = \|T(\alpha \hat{v})\| \leq C\|\alpha \hat{v}\|$ for $v = ...
2
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1answer
30 views

Why is the Span of a subset of a linear space defined in such at way?

If I have a subset $M$ of a linear space $E$, we define the linear span of the subset, $M$, as: $$\operatorname{span} M=\bigcap_\alpha \{E_\alpha : E_\alpha \hookrightarrow E\text{ and } M \subseteq ...
2
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2answers
58 views

If $A$ is a $4 \times 4$ matrix with $rank(A) = 1$, then either $A$ is diagonalizable or $A^2 = 0$, but not both

If $A$ is a $4 \times 4$ matrix with rank$(A) = 1$, then either $A$ is diagonalizable (over $C$) or $A^2 = 0$, but not both (Note that $A$ has complex entries) So far, the only thing I've tried ...
4
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1answer
54 views

Let the plane V be defined by $ax + by + cz + d = 0$; with $a, b, c, d \in \mathbb{R}$ and the vector $(a; b; c)$ a unit vector.

I am battling to get my mind around some of the concepts involving vectors in $3$-space. This question asks me whether the following statements are True or False: (A) The line $(a; b; c)$ is parallel ...
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1answer
48 views

$O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$

During a lecture of a Lie Algebras yesterday, the professor of the class stated the following fact without proof $O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$ Note that we are viewing ...
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1answer
24 views

parameterizing two points along a circular path [on hold]

Question: Parameterize points (3, 4) to (-4, 3) along a circular path I know that if I find the vector equation for this it would be r = <3-7t, 4-t>. But I'm not sure what the question is asking ...
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2answers
57 views

Let $A$ be a $5\times 5$ matrix all of whose eigenvalues are zero. Is $A$ symmetric, anti-symmetric, or $A=-A$?

Let $A$ be a $5\times 5$ matrix all of whose eigenvalues are zero. Which of the following are always true: a. $A=-A$ b. $A^t=-A$ (anti-symmetric) c. $A^t=A$ (symmetric) d. $A^5=0$ For b: ...
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3answers
38 views

Let $A\in \mathbb C$ be a $2 \times 2$ matrix, let $f(x)=a_0+a_1x+\cdots a_nx^n$ be any polynomial over $\mathbb C$. Comment on $f(A)$

Let $A\in \mathbb C$ be a $2 \times 2$ matrix, let $f(x)=a_0+a_1x+a_2x^2+\cdots a_nx^n$ be any polynomial over $ \mathbb C$. Then which of the following is true? a) $f(A)$ can be written as ...
2
votes
1answer
25 views

Generator operator of $v$?

Let $\text{U}_+$ be an associative $\mathbb{C}$-algebra with two generators $E$, $H$, and one defining relation $HE - EH = 2E$. Let $M$ be an $\text{U}_+$-module. If $v \in M$ is a nonzero eigenvector ...
0
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0answers
10 views

Why quadratic forms with the same isotropic cone are proportional?

Let $V$ be a complex space with a quadratic forms $P$, $Q$. Assume that the isotropic cones for $P$ and $Q$ are the same: $$ P^{-1}(0)=Q^{-1}(0). $$ How to check that there is a constant $c\neq 0$ ...
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0answers
35 views

Diagonalization: Differential Equations

The booking being used for this course is Differential Equations and Dynamical Systems by Lawrence Perko. The problem is as follows: Let the $n\times n$ matrix $A$ have real, distinct ...
0
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1answer
31 views

using Cayley Hamilton to find a power of linear transformation

I have reached the the following formula using Cayley Hamilton: $$T^3-2T+2I=0$$ Now I need to find $T^4$ so what I did is $T(T^3-2T+2I=0)\iff T^4-2T^2+2IT=0\iff T^4=2T^2-2IT$ But the answer is ...