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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39 views

Retrieve the value of x,z and x [on hold]

I want to learn about HOW to calculation in order to retrieve the value of x, y and x. Do you have a recommended tutorial to for a beginner in relation to linear algebra in this specific case? I ...
-1
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0answers
28 views

Eigen vectors of a matrix multiplied with its transpose [on hold]

Do the eigen vectors of $A A^T$ and $AA^T$ belong to the row, column, null or left null spaces of the matrix $A$?
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1answer
39 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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1answer
43 views

The Calculation Process

I don't understand HOW the calculation is done to retrieve the value 9, -6 and 18? Thanks!
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0answers
12 views

on triangular matrices and inverses [duplicate]

Suppose we have $A$, an upper triangular matrix. Can we conclude that $A^{-1}$ must be upper triangular as well? $A$ non singular. I mean it seems obvious but how can we prove it? by induction?
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1answer
16 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 ...
1
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1answer
36 views

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\\-2\sin2t\end{bmatrix} + ...
2
votes
3answers
95 views

Proving a Set is a Vector Space [on hold]

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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0answers
19 views

What is connected components of pseudospectra of matrix polynomial? . [on hold]

What is connected components of pseudospectra of matrix polynomial? Please see this link
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1answer
23 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 ...
2
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1answer
37 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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31 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
votes
1answer
24 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 ...
-3
votes
3answers
46 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? [on hold]

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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1answer
31 views

How Fourier transform relates to interpolation space.

This refers to the link : http://en.wikipedia.org/wiki/Interpolation_space where in the History section it mentions that: "Many methods were designed to generate such spaces of functions, including ...
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1answer
311 views

Edge weight function for graph instance of scheduling and allocation problem

I have difficulties developing a proper (non-scalar) edge cost function $c_e$ for my resource scheduling problem, which I mapped into a graph problem. Processes $P_i$ need resources $R_i \in ...
1
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2answers
42 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 ...
2
votes
2answers
61 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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0answers
16 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 ...
2
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1answer
29 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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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 = ...
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0answers
56 views

Matrix column-wise multiplication operator

I'm trying to find the proper operator for a column wise multiplication. Consider $v=[v_1, v_2, ..., v_n]^T$ and $$A=\begin{bmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\a_{2,1} & a_{2,2} & ...
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2answers
45 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 ...
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2answers
58 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 ...
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2answers
27 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 ...
3
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1answer
38 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. ...
4
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3answers
73 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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1answer
35 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 ...
2
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0answers
28 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 ...
4
votes
1answer
56 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
60 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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2answers
47 views

Diagonalization of Matrix

I have a problem that says to find the values of k for which the matrix $A$ is NOT diagonalizable over $\mathbb{C}$. I know that I need to find the zeros for the characteristic polynomial and check ...
2
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2answers
10k views

How to find the absolute value of a vector?

In my linear algebra course I keep seeing something like this: a = {1, 3, 5} Then in formulas I see this: |a| What does this mean, what is the absolute value of a vector? Wouldn't just be {1,3,5}? ...
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0answers
7 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
11 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 $n-2$, ...
28
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1answer
2k views

Cauchy's Integral Formula for Cayley-Hamilton Theorem

I'm just working through Conway's book on complex analysis and I stumbled across this lovely exercise: Use Cauchy's Integral Formula to prove the Cayley-Hamilton Theorem: If $A$ is an $n \times n$ ...
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0answers
19 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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2answers
628 views

Largest eigenvalue of a symmetric positive definite matrix with rank-one updates

I have a $n \times n$ symmetric positive definite matrix $A$ which I will repeatedly update using two consecutive rank-one updates of the form $A' = A + e_j u^T +u e_j^T$ where $\{e_i: 1 \leq i \leq ...
2
votes
1answer
24 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 ...
0
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4answers
58 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 ...
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1answer
46 views

How do I prove $\lVert{x}\rVert_2\leq{1}$

If $x^Ty\leq1$ for all $y$ with $\lVert{y}\rVert_2=1$, then $\lVert{x}\rVert_2\leq{1}$. $x,y\in R^n$ I have tried to prove it by using the definition of vector inner product: ...
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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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0answers
37 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 ...
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2answers
20 views

$W$ 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 ...
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0answers
6 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.
2
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1answer
28 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 ...
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1answer
71 views
+50

A question about an infinite sequence of elementary row operations

Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$? "What can we multiply the top equation by ...
2
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1answer
33 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 ...
3
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2answers
62 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 ...
1
vote
2answers
60 views

calculating the characteristic polynomial

I have the following matrix: $$A=\begin{pmatrix} -9 & 7 & 4 \\ -9 & 7 & 5\\ -8 & 6 & 2 \end{pmatrix}$$ And I need to find the characteristic polynomial so I use ...