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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If $A$ is a square matrix and $Ax = b$ has a unique solution for some $b$, is $A$ necessarily invertible?

Let $A$ be a square matrix. Suppose that $A x = b$ has a unique solution for some $b$. Is $A$ necessarily invertible? I said no because the invertible matrix theorem states that $A x = b$ has a ...
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1answer
48 views

Residue class ring $\mathbb{Z}[x]$/I and $\mathbb{Z}[x]$/J

$I = \left\lbrace \sum_{i=1}^{n} a_ix^i : n \in \mathbb{N}, a_1, ..., a_n \in \mathbb{Z} \right\rbrace$ beeing an ideal of $\mathbb{Z}[x]$ with polynomials without a constant term and $J = \left\...
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Given A,B such that rank $A_{m \times n} = m \Rightarrow \exists C: AC = B$

Let A,B be matrices $m \times n$ and rank A $ = m$. How to prove that there exists matrix C such that AC = B?
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A matrix representation for the inverse matrix.

I have a problem from "Methods of Algebriac Geometry in Control Theory by Peter Falb" textbook: Show that if $A$ is $\,n\times n\,$ matrix, then $\displaystyle\,(zI-A)^{-1} = \sum_{j=1}^n \phi_j\...
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1answer
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Find eigenvalues of a matrix using Perron–Frobenius theorem

I have to find the largest eigenvalue of a matrix containing only positive entries: $$\left( \begin{array}{ccc} e^{a} & 1 & e^{-a} \\ 1 & 1 & 1 \\ e^{-a} & 1 & e^{a} \end{...
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1answer
32 views

Properties of difference of projections

Let A be real square matrix of order $\geq 6$ and $B = I - 2A$. Which of the following claims is (are) true? If A is a projection, then B is orthogonal. If A is a projection, then B is non-singular. ...
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1answer
40 views

Problem on symmetric matrices

Let $A$ be square non-singular matrix of order $n \geq 2$. If $A$ is symmetric, then $A^2$ is symmetric positive definite. If $A^2$ is symmetric positive definite, then $A$ is symmetric. I think I ...
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37 views

Change of coordinates matrix

My linear algebra textbook has an example that I don't really understand. We're trying to find the change of coordinates matrix from $B$ to $B'$ where $B = \{(1, 1), (1, -1)\}$ and $B' = \{(2, 4), (3, ...
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1answer
15 views

Linear map and skalar multiplication

Let assume linear map $L: \mathbb{C}^3 \rightarrow \mathbb{C}^3$, defined as $Lu:=\langle v,u\rangle v$ where $u\in \mathbb{C}^3$ and $v\in \mathbb{C}^3$ is non-zero chosen vector with its norm $||v||...
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Geometric mean of k smallest eigenvalues.

This problem comes from S. Boyd's Convex Optimization, problem 3.26(b). It states that the geometric mean of $k$ smallest eigenvalues of a positive definite matrix $X$ satisfies the following ...
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1answer
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simultaneous linear equations

If a pair of linear equations have infinite solutions, then does that mean they are consistent? And first of all, what does it mean if two simultaneous equations are consistent?
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1answer
117 views

Recursive Hexagon Problem to find number of hexagons at each stage

The source of this problem is this SPOJ question. Let me simplify it: A valid beehive is recursively defined as follows: 1. A single regular hexagon is a valid beehive. 2. To all the external cells of ...
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1answer
59 views

Can a set of integers be linearly indepedent over rational field $\mathbb{Q}$?

As title says, can a set of integers be linearly independent over rational field $\mathbb{Q}$ or integer ring $\mathbb{Z}$?
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1answer
41 views

Prove that $a^2 = b^2 \rightarrow (a = b $ or $ a = -b )$, where $a,b \in C $ and C is a field

Problem C is a field and $a,b \ \epsilon \ C$ Prove: $a^2 = b^2 \rightarrow a = b \vee a = -b $ How would I go about proving this. I'm not entirely sure, what this implies. I know that for ...
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0answers
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Is there a mathematical way to fold a $20 dollar bill for compactness?

I had a strange thought. I used to carry a pill fob on my keys with an emergency $20 bill in it, before the whole thing got stolen. I always had some trouble fitting the bill inside the fob and ...
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2answers
64 views

How is it distinguished in matrix multiplication which is the vector and which is the matrix representing a linear transformation?

The terminology that is used everywhere when applying a matrix to a "vector" is considered is this: the matrix represents a linear transformation and there is a row or column vector. But a matrix can ...
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3answers
210 views

How many degrees of freedom would a rotation matrix in R5 have?

I understand that a rotation matrix in R3 has three degrees of freedom because there is three linearly independent planes that the rotation can take place in. How does this translate to R5?
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1answer
33 views

Least square approximation where X vector is given.

I want to find a curve of the form $y = a + b \sqrt{x}$ that best fits the points: $(3, 1.5)$, $(7, 2.5)$ and $(10, 3)$ by substituting the $x$ vector $= \sqrt{x}$ My understanding of the process to ...
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1answer
109 views

Consider I'm a 10 year old kid, explain what “linearly independent” and “basis” means [closed]

As the question states. Consider I am a child, explain what those concepts mean.
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1answer
45 views

Where am I going wrong with finding eigenvectors?

Simple example, but I am having the same issues with all of the problems I attempt. $$A= \left[\begin{array}{rrr|r} 6 & 3 \\ 2 & 7\\ \end{array}\right] $$ I get eigenvalues 4 ...
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0answers
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Find a matrix to represent the mapping of a factor module

I have a problem from my past paper I can't figure the logic to, even after seeing the answers. The question goes 【Q】Let $V=\mathbb{R}[X]_{<4}$ be the vector space of real polynomials of degree ...
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1answer
131 views

Using SVD to approximate matrix-vector multiplication?

Given some matrix A, is it possible to use Singular Value Decomposition to approximate Ax for some vector x within some error bound? According to Efficient low rank matrix-vector multiplication, it ...
3
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1answer
157 views

Definition of Distinct eigenvalue clarification?

I'm solving a problem where I am given the eigenvalues of a matrix $A$ and need to solve for the determinant of $A$. I know that if my matrix is diagonalizable I can find the determinant of $A$ by ...
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1answer
42 views

Why is this valid $tr(VDV^{-1}) = tr(VV^{-1}D)$?

Given a diagonalizable matrix A, such that this relation holds: $A = VDV^{-1} $, where D is a diagonal matrix. Now the following relation is given when taking the trace: $tr(VDV^{-1}) = tr(VV^{-1}D) =...
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Gauss-Jordan Algorithm underdetermined equation system

$Ax=b$ with $A$ more columns than rows. If I apply Gauß Jordan algorithm, I get a diagonal matrix, where I can read off one solution. But don't I loose some solutions, because there are many ...
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1answer
47 views

Elementary Transformation

$Ax=b$ with $A$ integer Matrix and $b$ integer vector. Looking for solutions $x_i$ in $\mathbb{Z}$. So if we multiply by elementary transformation matrices: (Add an integer multiply of one column/...
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1answer
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Find the best approximation (in $L^2$ / mean-square sense) for $ln x$

The full statement of the problem is: Consider the set of two functions $\{1,x\}$ on the interval $x\in[0,1]$. Replace the second function by another one in $span\{1,x\}$ which turns the pair into an ...
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1answer
42 views

Similarly Commuting Matrices

I was wondering, if $D$ and $A$ are similar matrices, over $\mathbb{R,C}$, that is $D=S^{-1}AS$ and $DC=CD$, for some $C$, must $A$ commute with $C$? For some reason, this one is slipping from me. I ...
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4answers
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Suppose T is a linear transformation such that T(1,1,1)=(0,1,2), T(1,0,1)=(1,1,1), and T(0,0,1)=(1,2,3).

What is T(x,y,z) What I tried: T(x,y,z)= $$ \begin{pmatrix} 1&1&1|0&1&2\\ 1&0&1|1&1&1\\ 0&0&1|1&2&3\\ \end{pmatrix} $$ Which reduces to \begin{...
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2answers
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Well-defined and Equivalence relations

I am wondering why the following is well-defined... The definition of well-defined is given as; $g:(X/\sim) \to Z$ is well-defined if a mapping $f:X \to Z$ can be found where $f$ has the property $x \...
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1answer
25 views

Use separation of variables to find a solution $u= u(x,t)$…

So I get up to the last paragraph of the solution. I can get the bases of the solution, but beyond that, I'm really confused as to what they did. Any help would be appreciated!
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1answer
38 views

Have a question about Linear Transformations

Explain why there cannot be a linear transformation T: $R^2$ --> $R^2$ for which T(1,1)=(2,3) and T(3,3)=(1,4). I have no clue how to start this problem. Wouldn't $T(4,4)$=$4T(1,1)$=$4(2,3)$=$(8,12)$...
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1answer
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If the entries of an invertible matrix N are between -1 and 1, is its operator norm less than 1?

For Euclidean norm. If so, why? If not, might $(I-N)^{-1}$ exist some other way? This spins-off from here.
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4answers
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Prove that $ A = - A^{\top} $ and $ \text{rank}(A) \leq 1 $ imply $ A = \mathbf{0} $. [closed]

Let $ A \in {\text{M}_{n}}(\Bbb{R}) $, and suppose that we have the following: $ A = - A^{\top} $. $ \text{rank}(A) \leq 1 $. Why then is $ A = \mathbf{0} $? Thanks!
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linear algebra bound

I have a problem in course project? I have two positive definite $n\times n$ matrices, $A$ and $B$. I want to find the bound of singular values of a product of these matrices ? these matrices are ...
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0answers
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Find all $(x,y,z)\in \Bbb Z^3$ so that $ x^y=x(\text{mod }z)$

I got this problem for Integer Triplets I tried this way : $ x^y=x(\text{mod }z)$ which implies $ x^y-x=0(\text{mod }z)$ and then i got stuck, What should I do? is their a better way
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1answer
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Theorem** on page 288 from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

1Let $V$ a $2n$ complex vector space and take on $V$ a quadratic form. Now define $$ \Sigma=\{\Lambda:Q(\Lambda,\Lambda)\equiv0 \} \subset Gr(n,2n)$$ where $\Lambda$ is a maximal subspace i.e it is ...
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2answers
110 views

How to show $f(x)$ has no root within $\Bbb Q$

A polynomial problem from my old algebra textbook: $f(x)\in\Bbb Z[x]$ with leading coefficient $1$, $\deg f(x)\ge 1$, and both $f(0)$ and $f(1)$ are odd numbers, prove: $f(x)$ has no root within ...
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1answer
42 views

What seems to be the minors of the Adjugate matrix $\text{adj}(A)$ of a square matrix $A$?

It is by definition that entries of the adjugate matrix $\text{adj}(A)$ are the corresponding $(n-1)$-minors of $A$ (up to a sign). What can we say about the $k$-minor of $\text{adj}(A)$ in relation ...
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1answer
49 views

Eigenvector of a $C^n$ class matrix

Let $A$ be the following matrix function: $\Bbb{R} \to \Bbb{R}^{a \times (a+1)}$ $t \mapsto A(t)$ Let us suppose that $A$ is $C^{\infty}$, meaning that all of $A$ coefficients are $C^{\infty}$. Let ...
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1answer
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Theorem 3.1 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: Is the notion of convex set valid in complex vector spaces?

Is the notion of convex sets valid and meaningful for complex vector spaces? Or, do we need to restrict ourselves to real vector spaces and normed spaces when we talk about convex sets? The ...
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1answer
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$A$ is singular and normal matrix, what must be its characteristic polynomial?

Let $A$ be a $5\times5$ real singular matrix which is normal. If $1-2i$ is an eigenvalue of $A$ and $2+i$ is an eigenvalue of $A^*$ (conjugate transpose), what must be its characteristic polynomial? ...
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1answer
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Do matrices have average and fluctuations?

Given a set of numbers, one can calculate the average of those numbers and the fluctuation (variance) over the average. E.g,, $\langle A \rangle=\frac{1}{N}\sum_{i=0}^N A_i$ and $(\delta A)^2 = \...
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1answer
46 views

Field extensions and algebraic elements

Can somebody explain why taking beta gives $K(\beta)$ as a subspace of $K(\alpha)$?
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2answers
121 views

Naturality in linear algebra

Question. How can we formalize these intuitions about predicates on matrices? Let $P$ denote a predicate on matrices, so that $P(A)$ is true for some choices of matrix $A$ and false for all others. ...
4
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1answer
89 views

Linear decomposition of positive semi-definite matrices

It is true that the vector space of $n\times n$ Hermitian matrices is an $n^2-$dimensional real vector space and that one can find a basis for this space consisting exclusively of positive semi-...
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Compute the limit of a matrix

We need to compute the limit of a sequence as $x \rightarrow \infty$ Using matrix-matrix multiplication we can define power as $A^p=A*\cdots*A$, $p$ times. We need to compute the limit of $A^p$ as $...
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Least square matrix form will fail, if the inverse property not satisfied?

In the matrix form of least squares , the inverse of ( X transpose X ) we are calculating . So, what if that matrix does not posses inverse properties. I mean what if it is not invertible ? Sorry if ...
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2answers
98 views

Under what conditions does $(I-N)^{-1}$ exist?

Given an nxn matrix N and $I=I_n$, under what conditions does $(I-N)^{-1}$ exist? On one hand $(I-N)(I + N + N^2 + ...) = (I + N + N^2 + ...) - (N + N^2 + ...) = I?$ On the other hand, $(I-N)(I + N +...
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1answer
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What is the least positive integer $m$ such that $\text{rank} A^m=\text{rank} A^{m+1}$? [closed]

Let $A$ be a complex square matrix. What is the least positive integer $m$ such that $\text{rank} A^m=\text{rank} A^{m+1}$? Express $m$ in terms of some quantities associated with $A$.