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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Solution space for a set??

Make a system with 3 eqns and 3 variables of which the solution space is spanned by V =[ 1 3 0], [1 0 -1]? Do I do the cross product of these two? Can someone show me how to do this? Also Can you ...
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9 views

Resolvent matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$? Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
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2answers
31 views

Find the eigenvalues of the following matrix

Consider $A =\left( \begin{array}{ccc} -1 & 2 & 2\\ 2 & 2 & -1\\ 2 & -1 & 2\\ \end{array} \right)$. Find the eigenvalues of $A$. So I know the characteristic polynomial is: ...
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3answers
34 views

Prove $A_1, A_2, \ldots,A_{\ell-1}, A_{\ell+1}\ldots \ldots, A_{m} $ in $\mathbb R^m$ are linearly independent viewed as vectors in $\mathbb R^{m-1}$?

Suppose I have linearly independent vectors $A_1, A_2, \ldots, A_m$ in $\mathbb R^m$ Consider the matrix $B = [A_1, A_2, \ldots A_m]$ consisting of these vectors and suppose $B^{-1}[A_1, A_2, ...
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31 views

Prove a certain matrix is positive semidefinte.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi$ to be a diagonal matrix with a principal left eigenvector of $P$ on the main diagonal and zeros ...
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16 views

Given multiple polynomial equations find a basis.

I have read several other threads on Math.SE, including the similarly titled: basis of the polynomial vector space I've also checked out a video lecture on Youtube by njwildberger, but I simply have ...
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13 views

Analytical solution to the first PCA direction

It is known that the first PCA direction for a dataset of $n$ points is the unit vector with max variance after projecting the points onto this vector. I wonder whether there are some analytical ...
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0answers
17 views

A question on vectors represented by multilinear polynomials

Let the set of multilinear polynomials of degree atmost $t$ in $\Bbb Z[x_1,\dots,x_n]$ be $\Bbb Z^{t}[x_1,\dots,x_n]$. Let $S=\{0,1\}^n$. Fix an ordering of $S$. For every $f\in\Bbb ...
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0answers
8 views

Uniqueness of k-sparse solution in compressed sensing

This question has been bothering me for some time now. Please correct me if I've got anything wrong. In compressed sensing, we try to find a unique k-sparse solution to an underdetermined system by ...
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1answer
23 views

“a matrix is positive semi-definite” not necessarily equavalent to “all leading principle minors are nonegative”?

Have a look at this matrix: $$ H = \left( {\begin{array}{*{20}{c}} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{array}} \right).$$ All the leading ...
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1answer
28 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
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27 views

Lyapunov function

How to do this problem? Find a Lyapunov function for $(0,0)$ in the system: $$x˙=3xy^2−11x^2$$ $$y˙=11x^3−4y^3$$ I know there is no formula for finding Lyapunov functions for a system, so how do I ...
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33 views

Prune a linearly independent set? What is the element of Span(Z)? What is the

Consider the following subset of $P_3(\mathbb{R})$ (real polynomial functions of degree at most 3). $$ Z = \{f_1, f_2, f_3, f_4, f_5\} $$ where $f_1(x) = 1-2x+2x^2-x3$, $f_2(x) = 1-x+x^2+x^3$, ...
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2answers
46 views
+50

Is the product of (modified) adjacency matrices of two matchings necessarily symmetric?

Consider $n$ vertices, and two (not necessarily perfect) matchings $M_1$ and $M_2$. With the following definition of a (modified) adjacency matrix of a matching, can we claim that $A(M_1)\cdot A(M_2)$ ...
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37 views

Find the determinants of the given matrices

Consider scalars $a,b,c,d,e,f$ such that $\det\left( \begin{array}{ccc} a & 1 & d\\ b & 1 & e\\ c & 1 & f\\ \end{array} \right) = 7$ and $\det\left( \begin{array}{ccc} a & ...
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21 views

Linear algebra eigenvalue proof

I asked this question before but after thinking about it I was trying to think of another way. Proof multiplied complex matrix has non negative eigenvalues I need to show given that A is a matrix $\in ...
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3answers
37 views

Formulation of the matrix determinant

How was the idea (and the equation) for the determinant of a square matrix formulated, and why does it work? All I've learned is that the determinant of a matrix is 0 when some row is a linear ...
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1answer
16 views

Proof multiplied complex matrix has non negative eigenvalues

I need help to show given that $\mathbf A$ is a complex nxn matrix that $\mathbf {AA}$* is a Hermitian matrix and the eigenvalues > 0 Where * is taking the transpose of the complex conjugate of the ...
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Do prove in vector space (about span and subspace)

(a) let vector u = (a,b,b,0) and vector v = (0,c,-c,d) because u dot v = 0, thus v = c(0,1,-1,0) +d (0,0,0,1) therefore, W2 = span {(0,1,-1,0),(0,0,0,1)} the basis for w2 is (0,1,-1,0),(0,0,0,1) for ...
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dimension of vector space $\frac{\langle e_{ab_1\ldots b_p}\rangle}{\langle \sum_{1\leq i\leq p}e_{ab_1\ldots \widehat{b_i}\ldots b_pc}\rangle}$

Let $p$ be a prime and $n\!\in\!\mathbb{N}$. What is the dimension of the $\mathbb{Z}_p$-module $$V_{p,n}=\frac{\langle e_{ab_1\ldots b_p};\: 1\leq a<b_1<\ldots<b_p\leq n\rangle}{\langle ...
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16 views

Proving Equality of Hermitian matrix

I need to show that the equality $\ A \mathbf u \cdot \mathbf v = \mathbf u \cdot A\mathbf v \ $ holds true for all $\mathbf u, \mathbf v \in \mathbb{C^n}$ if and only if the matrix A is Hermitian ...
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1answer
465 views

Find an ordered basis $B$ for $M_{n\times n}(\mathbb{R})$ such that $[T]B$ is a diagonal matrix for $n > 2$

I have a homework problem that I'm stuck on. It is problem 5.1.17 in the Friedberg, Insel, and Spence Linear Algebra book for reference. "Let T be the linear operator on $M_{n\times n}(\mathbb{R})$ ...
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1answer
29 views

Every skew-symmetric matrix has a non-negative determinant

I'm breaking this up into the even case and odd case (if $A$ is an $n\times n$ skew-symmetric matrix). So when $n$ is odd, we have: $\det(A)=\det(A^T)=\det(-A)=(-1)^n\det(A)\Rightarrow ...
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1answer
16 views

This is only a subspace if $b=0$ - Axler - LADR p13

I have written here in Axler - Linear Algebra Done Right, page $13$. If $b\in \mathbb{F}$, then $\{(x_1,x_2,x_3,x_4)\in \mathbb{F}^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb{F}^4$ if and only ...
3
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5answers
212 views

Is the determinant of this matrix positive or negative?

$\left( \begin{array}{ccc} 1 & 1000 & 2 & 3 &4\\ 5 & 6 &7&1000 &8\\ 1000&9&8&7&6\\ 5 & 4&3&2&1000\\ 1&2&1000&3&4\\ ...
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0answers
13 views

Lovasz number and bipartite complement

Let $G=(V,E)$ be a graph on $n$ vertices. An ordered set of n unit vectors $U=(ui|i∈V)⊂R^N$ is called an orthonormal representation of G in $R^N$, if $u_i$ and $u_j$ are orthogonal whenever vertices i ...
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5answers
119 views

Rank of a matrix $A^2$ without calculating the square

I have a matrix $A=\begin{bmatrix} 2 & 0 & 4\\ 1 & -1 & 3\\ 2 & 1 & 3 \end{bmatrix} $ with rank 2. How do I prove that the matrix $A^2$ has also rank 2 without actually ...
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1answer
14 views

Basis and dimension of the subspace of solutions to $A\mathbf{x}=\mathbf{0}$

Consider $$ A =\left( \begin{matrix} 1 & -1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ \end{matrix} \right) $$ and find a basis and the dimension of $S(A,0)$, where $S(A,0)$ is the ...
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18 views

Link between the projection onto a subspace and projection onto hyperplane

The projection onto a hyperplane $H=\{x\in \mathbb{R}^n|\langle a,x\rangle=b\}$ is defined to be $P_{H}(x)=x-\frac{\langle a,x\rangle-b}{||a||^2}a,$ and characterized by $\langle c-p,x-p\rangle\leq0.$ ...
0
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1answer
21 views

Vector space generated

Let $(L,+)<(\mathbb R^n,+)$ be a additive subgroup and let $\{v_1,\ldots,v_m\}$ be a maximal linearly independent subset of $L$. Let $V$ be the subspace spanned by $\{v_1,\ldots,v_m\}$. Asumme that ...
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0answers
10 views

Double circle representation using ideal rubber bands

Let $V$ be a 3-connected planar graph, $V^*$ be its dual graph. Let $(C_i : i \in V)$ and $(D_p : p \in V^*)$ be a collection of circles so that if any edge $\{i,j\}$ borders any faces $a$ and $b$, ...
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2answers
34 views

Proof that every subspace is a vector space

I was unable to find a simple proof that a subspace is a vector space. I know that a subspace $S$ is a subset of a vector space, such that: $$\vec 0 \in S\\\vec a + \vec b \in S\\\alpha\vec a \in S$$ ...
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1answer
53 views

Equation. Proof that there is no solution.

Prove that $\nexists x,y \in \mathbb{N} $ such that: $$\begin{cases} x -y - 3k -1 = 0 \\ x-y -4l -2 =0 \\ x+y - 3f - 2 =0\\ x + y - 5m - 2 = 0 \\ x,y \in \mathbb{N} \end{cases} $$ I'm asking form any ...
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1answer
25 views

Find the bases of the intersection of kernel and image

Let $L_A$ :$R^{3}_{col}$ $\rightarrow$ $R^{3}_{col}$ , X $\rightarrow$ AX be operator of left multiplication by matrix $$A=\begin{bmatrix}1 & 2 & 1 \\ 0 & 1 & 1 \\ -1 & 3 ...
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Prove that $L(V,W)$ is a vector space over $K$

Let $V$ and $W$ are $k$-vector spaces and let $L(V,W)$ be the set of linear maps $V\to W$. I have to prove that $L(V,W)$ is a vector space over $K$. I have already done a step by step proof ...
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1answer
23 views

Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$.

Let $H=L^2(a,b)$ with $a<b$. Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$. Verify also that $$\{(b_1-a_1)^{-\frac{1}{2}} \cdot \dots ...
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1answer
26 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
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51 views

Question on matrix exponential

Let $A$ be a real matrix with real eigenvalues $\lambda_k$ and complex eigenvalues $\alpha_ k \pm i\omega_ k$ , all of which are simple. I'm trying to show that every element of the matrix $e^ {tA}$ ...
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1answer
20 views

Show matrix is element in eigenspace

Let $A$ be an $n\times n$ matrix such that $A^2=A$. a) Let $E_{1}(A)=\{x \in \mathbb{R^n} | Ax=x \}$: let $E_{0}(A)=\{{ x \in \mathbb{R^n} | Ax=0\}}$. Let $x$ be any vector in $\mathbb{R^n}$. Show ...
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1answer
18 views

Spectral decomposition - generalized eigenspaces

Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i ...
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0answers
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If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$

If $n$ is even, every skew-symmetric $n\times n$ matrix $A$ can be factored as $A=SBS^T$ where $S$ is a invertible matrix and $B$ has the form $B = \left( \begin{array}{ccc} 0 & a_1 & 0 & ...
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4 views

Find coordinates of point that satisfy given conditions

I have A(1,2,3) , B(-1,0,1), C(1,-1,1) which are points in $\mathbb{R}^3$. I'm trying to find another point H such that AH${\parallel}$AC and BH${\perp}$AC. I set H = ($h_1$, $h_2$, $h_3$), and took ...
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Eigendecomposition Parameterization of Real Matrix

Given a set of distinct non-real eigenvalues $\lambda_1, \dots, \lambda_N$, so that $\lambda_{2n} = \overline{\lambda_{2n+1}}$. Accordingly given a set of non-real orthonormal eigenvectors $v_1, ...
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1answer
17 views

Help solving system of linear equations.

In the process of running through an algorithm, I have derived the following systems of equations: i) $1/3 + 1/3x_1 + 1/3 x_6 = x_5$ ii) $1/2 + 1/4 x_6 = x_1$ iii) $1/2 + 1/2 x_5 = x_6$ I've tried ...
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0answers
34 views

Orthogonal basis of $R^n$

I have to prove that if orthogonal basis of $\mathbb R^n$ containing only vectors which coordinates are $1$ or $-1$ exisists then $n \leqslant 2$ or $n$ is divisible by 4. It's obvious that n have to ...
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2answers
69 views

Linear Algebra SubSpace Test Example

I'm struggling to understand how to conduct a SubSpace test, one example that I have no clue how to do, is $V=\Bbb R^3$ and $W= \{(t,-t\sin(\pi/3),t\sqrt2): t\in\Bbb R\}.$ Can anyone explain to ...
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2answers
51 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
0
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2answers
29 views

Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis ...
2
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1answer
35 views

understanding the matrix transpose

Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear ...
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1answer
22 views

Which of the following are true?

I need to find which of the following are true? $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$ My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim ...