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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Characterizing $\text{PGL}_2(\mathbb F_p)$

Where can I find a description and proof of the basic structure of $\operatorname{PGL}_2(\mathbb{F}_p)$ (Number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
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Singular Jacobian in Newton's method

How can we prove that Newton's method for a non-linear system converges linearly (as opposed to quadratically) if the Jacobian is singular at the root? Is this related to being multiple roots at that ...
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Can the zero vector be an eigenvector for a matrix?

I was checking over my work on WolfRamAlpha, and it says one of my eigenvalues (this one with multiplicity 2), has an eigenvector of (0,0,0). How can the zero vector be an eigenvector?
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Can a $3\times3$ matrix have more than $3$ linear independent eigenvectors?

I understand you can do multiples of eigenvectors, but suppose they are a linear independent. Can there be more than $n$ for a $n\times n$ matrix?
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Parametric / vector question.

Question 10 [10 points] Let L be the line with parametric equations $$ x = −6−3t $$ $$ y = 6+3t $$ $$ z = −8+2t $$ Find the vector equation for a line that passes through the point P=(−1, 2, 3) and ...
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A basic doubt to compute exponential of a matrix

Given a matrix I want to evaluate $e^{A}$. The method suggested uses the taylor expansion. But, it is also written that the method works well if the largest and smallest eigen values are not well ...
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1answer
14 views

How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...
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Does this guassian elimination have a solution?

I was asked to find the following solutions using guassian elimination, but I was unsure of my answers since it became quite messy but the variables still somehow fit: $$\left[\begin{array}{ccc|c} ...
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13 views

The expected value of a random vector when the X_is are independent

$ \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cov}{cov} $ The components of a random vector $\mathbf{X} = [X_1, X_2, \ldots, X_N]^{\intercal}$ all have the same mean $E_X[X]$ and the same ...
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0
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1answer
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Prove True or false : If A and B are nxn invertible matrices and (AB)^2=A^2B^2, then AB=BA

This looks like it is false but the thing is I can't find a counter example for it.
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Fixed field of the subgroup of $\operatorname{Aut}_{K}{K(x)}$

This link explains $\operatorname{Aut}_{K}{K(x)}$. And I want to know how to solve two problems below in the Hungerford's Algebra textbook (p.256). $7.$ Let $G$ be the subset of ...
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Matrix Norm Division

Suppose $A=uv^*$ where $u$ is an $m$-vector and $v$ is an $n$-vector. For any $n$- vector $x$, we can bound $||Ax||_2$ as follows: $||Ax||_2 = ||uv^*x||_2=||u||_2|v^*x|\leq||u_2||||v||_2||x||_2$. ...
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1answer
17 views

Number of positive, negative eigenvalues and the number of sign changes in the determinants of the upper left submatrices of a symmetric matrix.

How do we prove that the number of sign changes in the sequence of the determinants of the upper-left matrices of a symmetric matrix $A$ corresponds to the number of positive and negative eigenvalues ...
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19 views

linear algebra find a line that intersects another line

question: Let L be the line with parametric equations x = 3+2t y = −5 z = −6−3t Find the vector equation for a line that passes through the point P=(−5, 5, −6) and intersects L at a point that is ...
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1answer
20 views

Sets of binary sequences

In my course on linear algebra we have recently introduced linear independent subsets of vector spaces. As an exercise I have been thinking about examples of infinite linearly independent sets and ...
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2answers
31 views

Does basis of eigenspace mean the same as eigenvectors?

If you have a 3x3 matrix, 2 eigenvalues (one with multiplicity 2) and now 2 eigenvectors, how do you find the basis for each eigenspace?
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18 views

Orthogonal Matrices and Similarity Transforms

Sorry I can't be more specific with the title. I really don't know what to call this and about 2 hours of Googling has yielded no results. All we are given: $U$ is $n\times n$ and orthogonal $Ax = ...
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Can the number of sign changes in a sequence of determinants tell us how many negative eigenvalues a symmetric matrix has?

From notes, I've gathered that given a symmetric matrix, the number of sign changes in its characteristic polynomial is equal to the number of positive eigenvalues of $A$. Proof: Let $p(x)$ be a ...
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1answer
17 views

Uniqueness of Thin QR Factorization.

Let $A \in \mathbb C^{m x n}$, have linearly independent columns. Show: If $A=QR$, where $Q \in \mathbb C^{m x n}$ satisfies $Q^*Q=I_n$ and $R$ is upper triangular with positive diagonal elements, ...
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1answer
30 views

Simultaneous function with three variables using subsititution method

Use any substitution method and solve the following equations: $$2x+5y+7z=86$$ $$3x+y+5z=60$$ $$x+4y+3z=54 $$ I used $x+4y+3z=54$ to make $x$ the subject $x=54-4y-3z$.
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1answer
12 views

Covariance matrix of Y when we have the covariance matrix of X

If the random vector $\mathbf{X}$ is transformed according to \begin{align*} Y_1 &= X_1\\ Y_2 &= X_1 + X_2 \end{align*} and has a covariance matrix $$ \mathbf{C}_X = ...
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2answers
33 views

Does the line $(2,1,1)+t(-3,1,5)$ live within the plane $31x+3y+18z=62$?

I have a doubt with this exercise: Have the plane $$31x+3y+18z=62$$ What is the distance between this plane and some line $(x,y,z) = (2,1,1) + t(-3,1,5)$ for some $t\in\mathbb{R}$? The ...
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Differences of grade between this three books

I have a simple question, I noticed these three books for my study, but I didn't understand the grade of these books because the names of the paragraphs are similar . 1) ...
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1answer
19 views

Rank of a special matrix

Say a $5\times 5$ matrix $$A = \left[ \begin{array}{ccc} 1&2&3&4&5\\ 6&7&8&9&10\\ 11&12&13&14&15\\ 16&17&18&19&20\\ ...
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2answers
25 views

How to show that $\| QA\|_2=\| A \|_2$ where $Q$ is unitary (for a matrix A)

I want to show that for a unitary matrix $Q$ and a matrix $A$ that $$ \|QA\|_2=\|A\|_2$$ I start with the definition of matrix induced norms: $$\| QA \|_2 = \sup_{x \neq ...
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How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?

How to show that the surjectivity of a linear map $f: R^ n\to R^n$ implies the injectivity and vise versa?
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2answers
25 views

Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
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2answers
22 views

Number of Jordan canonical form of a matrix

Let, $A\in M(3,C)$. Assume that the characteristic & minimal polynomial of $A$ are known. Then what is the number of possible Jordan form of $A$ and how? What changes if we replace $C$ by $R$ or ...
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Eigensystem of a real symmetric Toeplitz matrix of large order

My question is related to this one. I am looking for the eigenvalues and eigenvectors of a square, symmetric, real Toeplitz matrix of order N where N is large. There are some references in the above ...
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1answer
30 views

Are the following quotient spaces finite dimensional?

If we take $\mathbb{F}[x]$ to be the set of all polynomials over the field $\mathbb{F}$, $E$ to be the subset of all such even polynomials, $N$ to be the set of these polynomials that have degree less ...
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Generalize discrete Lyapunov equation for n-th order linear dynamics system

My specific application is analysis of dynamic textures using linear dynamics systems of the form $$ I(t) = Cz(t) + w(t) \\ z(t + 1) = Az(t) + Bv(t), $$ where $I(t)$ is the original signal, $z(t)$ ...
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1answer
27 views

A question on a nonnegative quadratic form

Denote $x,y,z$ as variables, and $a,b,c$ as coefficients. Suppose $a\leq b\leq 0\leq c$ and $a+b+c=0$. Could anyone help me prove whether the following quadratic form positive semi-definite? ...
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Finding the co-ordinate vector

I can find the co-ordinate vectors for all $x$ in $R^n$ but I can't wrap my head around the ones for $x$ in $P_n$. Here is a question: Let $V$ be the space $P_3$ of all polynomials of degree at ...
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1answer
14 views

What is the relationship between parallelogram law and polarisation identity?

According to wikipedia article on polarisation identity, in a normed space $(V, || . ||$), if the parallelogram law holds, then there is an inner product on V such that $||x||^2 = \langle x, x\rangle$ ...
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1answer
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Speed of two points on a circle

Problem Two points $A$ and $B$ are moving on a circle at constant speeds $v_A$ and $v_B$. We assume that they start from the same position and that they instantly accelerate to their final speed. ...
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1answer
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Proof: F is isometric if and only if its matrix is orthogonal/unitary

I'd like to show that $F \in End(V)$ isometric $<=> M_{\beta \beta} (F)$ orthogonal/unitary But it seems as if I still have some trouble doing that ;/ "=>" $<v_i, v_j> = ...
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In a vector space $V$, If $v^tAw = v^tBw$ for all $v,w \in V$, does it imply $A=B$?

OK. So, basically this question came up when I was trying to solve a homework question about how the matrix representation of a bi-linear form on $V$ changes if we change the basis on $V$. Let's say ...
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3answers
276 views

A method of finding the eigenvector that I don't fully understand

Let $$A=\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & t \\ \end{pmatrix}$$ Which has a known eigenvalue : $\lambda$ Find the corresponding eigenvector Over the ...
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1answer
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Help me understand Vector Spaces (proving linear spaces)

Please help me understand each part clearly. Please don't give general answers, it's easier for me to understand concepts by doing specific questions and learning about them. (i) The reason ...
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1answer
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By given equation, finding orthogonal projection

Find the orthogonal projection of line with direction vector $u = ( 1 , 2 , 0 )$ onto the plane described by equation $-3x - 2y + 2z = -2$ i have tried to search ...
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3answers
28 views

How to determine volume of parallelepiped by 4 points

Let points $(0,0,0), (1,2,x), (-2,1,0)$ and $(1,1,3)$ be at four corners of a parallelpiped. Determine the volume of the parallelepiped by using the determinant in terms of $x$. For what value of $x$ ...
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What does inner product actually mean?

What does inner product actually mean? So far most of the cases that I encounter seems to suggest that dot product is the only useful inner product. I mean most of the things that we discuss about ...
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Which curve we will get under $\mathcal{A} \in M_2(\Bbb{R})$ from a unit circle

If I have a circle: $x^2+y^2=1$, It's parametric equation is : $$\begin{cases} x = \cos\theta \\ y = \sin\theta \end{cases}$$ under some transform: $A=\begin{pmatrix} a & b\\ c & d ...
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Speeding up solving a linear system

I need to speed up calculating the following linear system: $$ (A^TA +\rho I + \nu \sum_{k=1}^l (q_{k,1}q_{k,2}^T+q_{k,2}q_{k,1}^T))x=b, $$ where $A\in\mathbf{R}^{m\times n}$, $\rho,\nu$ in ...
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51 views

determine signature of matrix

what is the signature of this matrix: $\begin{pmatrix} -3&0&-1 \\0&-3&0 \\ -1&0&-1 \end{pmatrix}$ ? I tried calculating them without eigenvalues; this should be done via ...
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1answer
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Why isn't the orthogonal vector to a direction vector of a plane not necessarily perpendicular to such plane?

I had made a question, and the problem with my exercise was that I was trying to calculate a vector perpendicular to some plane in $\mathbb{R}^3$: given one line $L$ inside the plane, I grabbed the ...
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3answers
55 views

Does there exists a vector v such that $Av\neq 0$ but $A^{2}v=0$

Let A be a $4\times4$ matrix over C such that $\operatorname{rank}A=2$ and $A^{3}=A^{2}\neq0$. Suppose that A is not diagonalizable. My question is , "Does there exists a vector $v$ such that ...
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How to create an equation from this problem?

A full cola bottle is $2. 2 caps can be exchanged with 1 full cola bottle. 4 empty bottles can be exchanged with 1 full cola bottle. If you have $20, how many full coke bottles you will totally ...
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Matrix Decompositions: Difference between Cholesky Decomposition, Eigendecomposition and Jordan Normal Form Decomposition

I recently created a related topic about the square root matrix, in case you'd like to refer to that one. Here's what we want: Consider the matrix $\Omega=E(\mathbf{u}^{\top}\mathbf{u})$, where ...