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.
82,560 questions
0
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3answers
47 views
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 ...
2
votes
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 $\...
0
votes
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 ...
0
votes
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 $x$...
2
votes
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 $\tau$ ...
-1
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
17 views
LU decomposition of a matrix similar to tridiagonal
Let A be a matrix $$A=\begin{bmatrix}
*& *& 0& 0& *\\
*& *& *& 0& 0\\
0& *& *& *& 0\\
0& 0& *& *& *\\
*& 0& 0& *&...
1
vote
0answers
16 views
Gaussian elimination in vector spaces
I've been working on a set of problems while learning matrix operations as well as vector spaces and subspaces. But now I have some doubts that go outside the general rule of thumb and I'm unable to ...
2
votes
0answers
36 views
Interesting applications of density to prove difficult theorems
If we wanted to prove certain statements for every element of a set $S$, a possible approach is to prove the statement for a certain dense subset $S'\subset S$ (with respect to a certain metric), then ...
0
votes
1answer
22 views
Prove that if $L_1, L_2 \in L (R^2, R^2)$ then $(L_1 \oplus L_2) \in L (R^2, R^2)$ also.
We are given two linear functions $L_1 L_2 \in L (\mathbb R^2, \mathbb R^2)$ which we define addition as $(L_1 \oplus L_2)$ to be a map $(L_1 \oplus L_2) : R^2 \to R^2$ given by the formula,
$$(L_1 ⊕ ...
0
votes
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}...
2
votes
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 $...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
3
votes
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?
1
vote
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 ...
0
votes
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 ...
0
votes
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 $\vec{x},\vec{b}\in\mathbb{R}^n$...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
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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 ...
0
votes
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?
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
11 views
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 $$A=\left ( \begin{matrix}
-1&...
0
votes
0answers
11 views
Find out all equivalence classes?
Two Integer symmetric matrixes $A$ and $B$ are called equivalent with each other, if there exists $γ \in SL(2,Z)$ such that $B = γ^{T}Aγ$.
Obviously, equivalent matrixes share the same determinant. ...
2
votes
1answer
28 views
How does a transformation matrix of linear map defined on square matrices look like?
I am quite confused how the transformation matrix looks like in standard basis when the linear map is defined on square matrices.
Given a mapping $T: M_2(\mathbb{R}) \rightarrow M_2(\mathbb{R}) $ ...
2
votes
2answers
27 views
Number of solutions of an equality with absolute value operator
Consider:
$$ \left|\left|\left|x-1\right|-2\right|-4\right|=4 $$
What is the number of solutions for this equation? This one was particularly easy to me. If first observed that if this inequality ...
0
votes
2answers
32 views
Determine value of '$a$' for which the system is inconsistent and has infinitely many solutions.
Consider the matrix $A$, to be equal to:
\begin{bmatrix}1&2&1\\-1&4&3\\2&-2&a\end{bmatrix}
Then we can rewrite this as:
\begin{bmatrix}1&2&1\\0&6&4\\2&-2&...
0
votes
3answers
23 views
Suppose $V$ is finite-dimensional with dim $V \gt 0$, and $W$ is infinite dimensional. Prove that $\mathcal{L}(V,W)$ is infinite dimensional.
Suppose $V$ is finite-dimensional with dim $V \gt 0$, and $W$ is infinite dimensional. Prove that $\mathcal{L}(V,W)$ is infinite dimensional. (The question is from Linear Algebra Done Right)
My ...
1
vote
0answers
35 views
Homs and Tensor products, a specific isomorphism?
Let $L$ be a finite field extension of $\mathbb Q_p$ and $\mathfrak o$ its ring of integers. Let $M$ be an $\mathfrak o$-module and $N$ a $\mathbb Z_p$-module. We then have the map $$ \Phi \colon \...
1
vote
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 ...
2
votes
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:
$$det(B-\lambda ...
1
vote
1answer
68 views
If $x'=Ax$ and $y'=Ay$, does it mean that $x=y$?
The textbook says that the statement $x = y$ is False for this situation. Though I can't seem to understand it.
I put my understanding of the process of solving a system of differential equations by ...
0
votes
1answer
15 views
Suppose a square matrix $A$ has spectral radius $\rho(A) < 1$. Fixing the last row and scaling other entries by $r \in (0,1)$, will $\rho(A)<1$?
Suppose $A \in M_n(\mathbb R)$ has spectral radius small than $1$, i.e., $\rho(A) < 1$. Denote $A = \pmatrix{a_1^T \\ \vdots \\ a_n^T}$, where $a_j^T$ denotes the $j^{th}$ row of $A$.Putting $B=\...
0
votes
0answers
17 views
Change of basis of a lattice
Let $\Gamma:=\Big\{(i, 0), (0,i), (\sqrt 3,\sqrt 3)\Big\}$ be a basis of a lattice in $\mathbb C^2$.
Can we find another basis $\Gamma':=\{(1,0),(0,1), w\}$ such that span over $\mathbb R$ of $\Gamma$...
1
vote
1answer
30 views
Determine all positive powers.
Determine all positive powers of the $5 \times 5$ matrix...
$$C=\begin{bmatrix}
0&0&0&0&0 \\
1&0&0&0&0 \\
0&1&0&0&0 \\
0&0&1&0&0 \\
0&...
0
votes
0answers
32 views
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/...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
