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Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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Using the product of the roots of $(z+1)^n=1$ to prove that $\prod_{k=1}^{n-1} \sin\frac{k\pi}{n}=\frac{n}{2^{n-1}}$

$z_1,z_2,…z_n$ satisfy the equation $(z+1)^n=1$. Use the product of $z_1,z_2,…z_n$ to prove that $$\sin \frac {\pi}{n} \sin \frac {2\pi}{n}…\sin \frac {(n-1)\pi}{n}=\frac {n}{2^{n-1}}$$ Attempt I ...
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1answer
21 views

Proving matrices equation when all the matrices in it may not be invertible

I'm reviewing linear algebra for my exams this year, and I just encountered this problem. For an arbitrary matrix, $\boldsymbol{A} \in \mathcal{R}^{m \times n}$, prove there must be a unique matrix $\...
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0answers
24 views

Systematic way to find a matrix that satisfies an equation

Background It is relatively easier to find a solution to a system of linear equations in the form of $A\textbf{v}=\textbf{b}$ given the matrix $A$. But what systematic ways are there that allows us ...
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0answers
10 views

Notation for the summation of the cumulative product of each row of a matrix

What steps should I take to achieve the desired outcome? Given a matrix $A = \left[\begin{array}{ccc} 2 & 3 & 5\\ 1 & 4 & 11\end{array}\right]$, the steps should result in a scalar $x$...
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0answers
20 views

Matrix anticommuting with four or five Dirac matrices

Consider the following four or five Dirac matrices\begin{gather}σ_{x,y}\otimes τ_0,σ_{x,z}\otimes\tau_z,\tag{*1}\\σ_{x,y}\otimes τ_0,σ_{x,y,z}\otimes τ_z,\tag{*2}\end{gather} where $\sigma$ and $\tau$ ...
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2answers
19 views

Which of the following sets of vectors span R^3?

Can anyone explain how to do this? Usually when the question gives a vector and asks whether it is in span with some other vectors I put them in a matrix and calculate the determinant and so on but ...
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2answers
33 views

Eigenvalues and Eigenvectors without matrix $A$

We guess that a $3\times 3$ matrix $A$ has eigenvalues $0, 3, 5$ with eigenvectors $u, v, w$. a) Find a base of $N(A)$, and a base of $R(A)$. b) Find a solution of the equation $Ax = v + w$. Find all ...
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0answers
26 views

What happen with eigenvalues when we change the matrix

Give an example to show that the eigenvalues change when we subtract multiple of a line fron another line. Give an explanation why if $0$ is one of the eigenvalues, then it doesn't change. I gave an ...
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0answers
17 views

LU decomposition of a matrix similar to tridiagonal

Let A be a matrix $$A=\begin{bmatrix} *& *& 0& 0& *\\ *& *& *& 0& 0\\ 0& *& *& *& 0\\ 0& 0& *& *& *\\ *& 0& 0& *&...
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0answers
16 views

Gaussian elimination in vector spaces

I've been working on a set of problems while learning matrix operations as well as vector spaces and subspaces. But now I have some doubts that go outside the general rule of thumb and I'm unable to ...
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0answers
36 views

Interesting applications of density to prove difficult theorems

If we wanted to prove certain statements for every element of a set $S$, a possible approach is to prove the statement for a certain dense subset $S'\subset S$ (with respect to a certain metric), then ...
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1answer
22 views

Prove that if $L_1, L_2 \in L (R^2, R^2)$ then $(L_1 \oplus L_2) \in L (R^2, R^2)$ also.

We are given two linear functions $L_1 L_2 \in L (\mathbb R^2, \mathbb R^2)$ which we define addition as $(L_1 \oplus L_2)$ to be a map $(L_1 \oplus L_2) : R^2 \to R^2$ given by the formula, $$(L_1 ⊕ ...
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0answers
21 views

Solving linear system of equations by block

I want to solve the following large system of linear equations: \begin{align} \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{c} x_{1} \\ x_{2}...
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1answer
16 views

(reference-request) Theorem of Frobenius regarding transforming integer-valued matrix in diagonal matrix with divisibility conditions

A contest paper cites the following theorem as a theorem of Frobenius: Let $A$ be a $m \times n$ integer-valued matrix. There exists a positive integer $r \leq \min(m,n)$ and two unimodular matrices $...
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1answer
28 views

Linear independence of gradient vectors

I have to find all feasible points that are regular for a function with constraints: $h_1(x_1,x_2,x_3) = 2x_1x_2+x_3^2=0$ $h_2(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2=4$ The definition says that if a ...
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1answer
37 views

Eigenvalues and Eigenvectors of special matrix $A=u u^T$

How to calculate the eigenvalues and eigenvectors of the special matrix $A=u u^T$, where $u \in R^n$. I wrote down the matrix, which is the linear combination of vector $u$ by itself, and clearly has ...
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1answer
17 views

When does the base of the image is directly comprized of the columns of a matrix?

I am confused with what methods and when to use them to find the base of the image of a matrix. Sometimes I see that they use gauss-jordan to find which columns have pivots and then they take the ...
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1answer
27 views

How $C[a, b]$ satisfies the axioms of vector space

My book states that the set of all continuous functions defined on a closed interval $[a, b]$ where $a ≠ b$ satisfies the axioms of a vector space. I am understanding that this means that for the ...
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1answer
31 views

Linear algebra identity evaluation

I really couldn't find anything related to this simple identity I came up with so: $$\vec{r}=(r_x,r_y)=(r_x, \angle0)+(r_y,\angle\frac{\pi}{2})$$ My thinking process was that $r_y$ is practically ...
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2answers
65 views

Groups, Rings and Fields.

I am asking for the analogy behind these structures names. Why is a "field" called a field? Is there an analogy between a usual ring (finger ring) and a mathematical ring?
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1answer
18 views

Gramian matrix after elementary matrix operations

Will a Gramian matrix remain a Gramian matrix after elementary matrix operations(for example, subtract a row from another row and similarly to columns)? Of course, we do this with symmetry about ...
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0answers
7 views

Distance between two points at same angle in trochoid curve

Anyone please help me to find out the distance in following case. Refer to the attached image. Consider an arbitrary point P on the circumference of a circle of radius r (mm). The point makes an ...
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0answers
18 views

Determining all possible ways to perform gaussian elimination on a linear system of equations

My first question is quite general: consider a linear system \begin{equation} A\vec{x}=\vec{b}, \end{equation} with $A\in\{-1,0,1\}^{n^2}$ being a full rank matrix and $\vec{x},\vec{b}\in\mathbb{R}^n$...
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0answers
15 views

How to prove that a cofactor of a matrix is $A$ is $(-1)^{i+j} \times $ a minor

Let $A \in M_n(\mathbb{R})$ : $A'_{ij}$ have the same columns as $A$ except the $j$th one which is a column full of zero except on the $i$th entry where it is a $1$. $A''_{ij}$ is the matrix ...
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0answers
29 views

Eigenvalues of the sum of commuting matrices

In a proof I am reading, the author states that given any commuting matrices $A$ and $B$, any eigenvalue of $A+B$ must be the sum of an eigenvalue of $A$ and an eigenvalue of $B$. Why is this true? Is ...
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1answer
37 views

The orthogonal complement of the orthogonal complement from “Linear Algebra Done Right”

The following content is from "Linear Algebra Done Right" by Sheldon Axler Corollary: Suppose $U$ is a finite-dimensional subspace of $V$. Then $$U = (U ^\perp)^\perp.$$ We need to prove the ...
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0answers
15 views

Merging two orthonormal bases without Gram-Schmidt

I have two sets of column vectors: $A = \{a_1,a_2,\dotsc,a_m\}$ and $B = \{b_1,b_2,\dotsc,b_n\}$. I have orthornormal basis for both of them individiaully. $\{u_1,\dotsc,u_p\}$ is an othornormal basis ...
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1answer
17 views

nilpotent endomorphism and $Im(f)+Ker(f) \neq dim(V)$

If I have an endomorphism between vector spaces $f:V \rightarrow V$, such that $Im(f)+Ker(f) \neq dim(V)$, is this equivalent to $f$ being nilpotent?
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1answer
27 views

How is the definition of linear independence for infinite sets useful?

Given this fact: "an infinite set is linearly independent iff all of its finite subsets are linearly independent", how can this help us determine whether an infinite set is linearly independent if ...
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0answers
25 views

How does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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2answers
33 views

Similarities between the idea of Group Homomorphisms and Linear Transformations

I am not sure if this question has been asked before, but my search did not return me any answers. While reading several online notes that attempt to give an intuitive understanding of group ...
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0answers
18 views

Proof that Irreducible Block Diagonally Dominant Matrix is Nonsingular

Let $A$ be an $n\times n$ block matrix with $ij$th block $A_{i,j}$, where each $A_{i,j}$ is a square matrix of the same size for all $i,j$. Assume that $A$ is block diagonally dominant: each of the ...
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0answers
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Application of linear algebra to encryption [on hold]

How can I solve the following exercise or a book reference where I can find this type of exercises? Consider the English alphabet is with space in white and the key $$A=\left ( \begin{matrix} -1&...
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0answers
11 views

Find out all equivalence classes?

Two Integer symmetric matrixes $A$ and $B$ are called equivalent with each other, if there exists $γ \in SL(2,Z)$ such that $B = γ^{T}Aγ$. Obviously, equivalent matrixes share the same determinant. ...
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1answer
28 views

How does a transformation matrix of linear map defined on square matrices look like?

I am quite confused how the transformation matrix looks like in standard basis when the linear map is defined on square matrices. Given a mapping $T: M_2(\mathbb{R}) \rightarrow M_2(\mathbb{R}) $ ...
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2answers
27 views

Number of solutions of an equality with absolute value operator

Consider: $$ \left|\left|\left|x-1\right|-2\right|-4\right|=4 $$ What is the number of solutions for this equation? This one was particularly easy to me. If first observed that if this inequality ...
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2answers
32 views

Determine value of '$a$' for which the system is inconsistent and has infinitely many solutions.

Consider the matrix $A$, to be equal to: \begin{bmatrix}1&2&1\\-1&4&3\\2&-2&a\end{bmatrix} Then we can rewrite this as: \begin{bmatrix}1&2&1\\0&6&4\\2&-2&...
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3answers
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Suppose $V$ is finite-dimensional with dim $V \gt 0$, and $W$ is infinite dimensional. Prove that $\mathcal{L}(V,W)$ is infinite dimensional.

Suppose $V$ is finite-dimensional with dim $V \gt 0$, and $W$ is infinite dimensional. Prove that $\mathcal{L}(V,W)$ is infinite dimensional. (The question is from Linear Algebra Done Right) My ...
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0answers
35 views

Homs and Tensor products, a specific isomorphism?

Let $L$ be a finite field extension of $\mathbb Q_p$ and $\mathfrak o$ its ring of integers. Let $M$ be an $\mathfrak o$-module and $N$ a $\mathbb Z_p$-module. We then have the map $$ \Phi \colon \...
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2answers
40 views

Existence of diagonalizable matrix

Let $A \in \mathbb{C}^{4 \times4}$ and $A^5 = I$. Is $A$ always diagonalizable? How to show $|\mbox{tr} (A)| \leq 4$? If $\mbox{tr} (A) = 4$, can we determine matrix $A$? Some ...
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1answer
39 views

Need help finding eigenvectors of matrix

I have the following Matrix: $$B= \begin{bmatrix} 8 & -6 & -6 \\ 30 & -22 & -30 \\ -30 & 30 & 38 \end{bmatrix} $$ I find the characteristic polynomial as: $$det(B-\lambda ...
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1answer
68 views

If $x'=Ax$ and $y'=Ay$, does it mean that $x=y$?

The textbook says that the statement $x = y$ is False for this situation. Though I can't seem to understand it. I put my understanding of the process of solving a system of differential equations by ...
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1answer
15 views

Suppose a square matrix $A$ has spectral radius $\rho(A) < 1$. Fixing the last row and scaling other entries by $r \in (0,1)$, will $\rho(A)<1$?

Suppose $A \in M_n(\mathbb R)$ has spectral radius small than $1$, i.e., $\rho(A) < 1$. Denote $A = \pmatrix{a_1^T \\ \vdots \\ a_n^T}$, where $a_j^T$ denotes the $j^{th}$ row of $A$.Putting $B=\...
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0answers
17 views

Change of basis of a lattice

Let $\Gamma:=\Big\{(i, 0), (0,i), (\sqrt 3,\sqrt 3)\Big\}$ be a basis of a lattice in $\mathbb C^2$. Can we find another basis $\Gamma':=\{(1,0),(0,1), w\}$ such that span over $\mathbb R$ of $\Gamma$...
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1answer
30 views

Determine all positive powers.

Determine all positive powers of the $5 \times 5$ matrix... $$C=\begin{bmatrix} 0&0&0&0&0 \\ 1&0&0&0&0 \\ 0&1&0&0&0 \\ 0&0&1&0&0 \\ 0&...
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0answers
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The definition of normal bundle as a quotient.

Let $f: N \to M$ be a smooth immersion and let $p \in M$, $W = f(V) \subset M$ be an submanifold with $q = f(p).$ Then the sequence is split exact $$T_qW \hookrightarrow T_qM \stackrel{\mu}\to T_qM/...
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2answers
52 views

How can we generalize the fact of finite dimensional vector space to an infinte dimensional case?

I am reading vector space from Friedberg. There in the last section they told about infinite dimensional vector space but there is not sufficient contents. Now my question is why can't we define ...
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0answers
10 views

Describe vector subspaces and FInd basis for sum of vector subspaces

I need help with this problem: For part a), I thought to write $v_1,v_2,v_3$ in their vector form with respect to the standard basis. Then I will group them together into a matrix and row reduce the ...
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0answers
14 views

Jacobi matrix with complex entries and optimization's orientation

What can we tell about the direction/ orientation of an optimization if the entries of the Nx1 Jacobi-matrix (gradiant) are complex? According to wikipedia: "if the Jacobian determinant at p is ...
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3answers
112 views

Intuition of generalized eigenvector.

I was trying to get an intuitive grasp about what the the generalized eigenvector intuitively is. I read this nice answer, so I understand that in the basis given by the generalized eigenvectors, a ...