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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5
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1answer
31 views

Subspaces of $\Bbb R^n$ containing vectors whose coordinates satisfy prescribed inequalities

For any integer $n\ge2$, the vector space $\Bbb R^n$ is divided into $n!$ "wedges" by prescribing which coordinate is largest, second-largest, etc. One such wedge is $$\{(x_1,\dots,x_n)\in\Bbb ...
1
vote
1answer
83 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
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0answers
22 views

One dimensional space $(\Lambda ^n V)^{1/2}$

Let $V$ be an $n$ dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ on $V\oplus V^*$ such that $(v+\xi , ...
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3answers
21 views

Find a matrix which maximizes expression

Assume I have column vectors $x,y\in\mathbb{R}^n$, and the following expression $$ A\in M_n(\mathbb{R}),\ |\det A|\leq 1,\ K(A) = x^tAy $$ How can I find the matrix $A$ that maximizes expression ...
4
votes
1answer
415 views

What are applications of Lagrange's identity?

I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then $$ \left|\sum_{i=1}^na_ib_i\right|^2 = ...
-4
votes
0answers
27 views

Proving linear algebra equation

I am having trouble proving that two multivariate formulas are equivalent. I implemented them in MATLAB and numerically they appear to be equivalent. I would appreciate any help on this. Prove A = ...
3
votes
1answer
33 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of several simplicies, with all coordinates being non-negative. That is, given $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to ...
3
votes
1answer
42 views

A Real Matrix, its Kernel and Image

This is an old exam problem: For an $m \times n$ real matrix $A$, define $\ker A = \{x \in \mathbb{R}^n \mid Ax=0 \}$ and $\operatorname{Im} A = \{Ax \mid x \in \mathbb{R}^n \}$. Show that for all $b ...
1
vote
1answer
26 views

Isomorphism between $E$ and $E^*$

Show that there does not exist a isomorphism $\phi:E\rightarrow E^*$ that it takes every basis of $E$ to its dual basis. ($E$ is a vector space over field K and $\text{dim}E=n$ .) My attempt: There ...
3
votes
1answer
43 views

Calculating the rank of a Boolean matrix and Boolean matrix factorization

I am interesting in some sort of algorithm for calculating the Boolean rank of small $M \times N$ Boolean matrices. Just to be clear, by Boolean matrices I mean matrices with entries $0$ or $1$ where ...
1
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3answers
1k views

Show that the diagonal entries of symmetric & idempotent matrix must be in [$0,1$]

Show that the diagonal entries of symmetric & idempotent matrix must be in [$0,1$]. Let $A$ be a symmetric and idempotent $n \times n$ matrix. By the definition of eigenvectors and since $A$ is ...
2
votes
0answers
16 views

Pullback maps and an equallity

Let $\Lambda^p (E)$ be the set of $p$-covariant exterior tensors on linear space $E$ over field $K$ (dim$E=n$ and $ 0\leq p\leq n $ , $\Lambda^0E:=K$). We define linear map ...
1
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1answer
29 views

Reformat this math formula to not need parenthesis

I've got a pricing equation that I am working with for an eCommerce site and I need to reformat this equation to not need parenthesis. Original Formula: {price} + ({length} * ({ppf} + ...
1
vote
1answer
28 views

Finding the standard basis for $\mathbb{R}^4$ that can be added to the set $\{(1,-4,2,-3),(-3,8,-4,6)\}$ to produce a basis for $\mathbb{R}^4$

Finding the standard basis for $\mathbb{R}^4$ that can be added to the set $\{(1,-4,2,-3),(-3,8,-4,6)\}$ to produce a basis for $\mathbb{R}^4$. I first check that the two vectors in the set are not ...
4
votes
3answers
409 views

Unusual result to the addition

Question: Prove that (666... to n digits)^2 + (888... to n digits)=(444... to 2n digits) My way: I just proved the given equation for three values of n and written at the bottom. "Since the ...
-1
votes
1answer
38 views
+50

Detecting linear dependencies in a matrix

Let $X$ be a matrix of $n$ rows (measurements) and $p$ columns (dimensions or features), and $n>p$. Denote by $r(i)$ the $i$th row of $X$. Assume that a subset of rows of $X$, denoted $r(i_j)$, ...
-2
votes
1answer
30 views

Orthogonal vectors and potential

given the potential $ψ(x;y)$, such that $dψ=−u_2dx+u_1dy$, why are $∇ψ=(−u_2;u_1)$ and $ψ(x;y)=c$ orthogonal vectors ? $c \in \mathbb{R}$ is a constant, and $\mathbf{u}(x; y) = (u_1(x;y); u_2(x;y))$, ...
0
votes
3answers
42 views

Finding the Characteristic Equation

For the following matrix I need to find $$\begin{bmatrix}-3 & 2 &1 \\3 & -4 & -3 \\-8 & 8 & 6 \end{bmatrix}$$ a. Characteristic Polynomial of $A$ b. Eigen Values c. Eigen ...
0
votes
0answers
20 views

Solving an specific equation involving cos and sin

Here is the equation: $|a|\sin(\alpha+2\theta)+|b|\sin(\beta+\theta)=0$, where $\theta$ is the variable, $a$ and $b$ are complex constant and their corresponding arguments are $\alpha$ and $\beta$. ...
2
votes
1answer
42 views

Desk Reference Question: Linear Algebra and its Applications by Lay or Strang? Or Handbook of Linear Algebra?

I would like a good working desk reference for linear algebra for someone in applied mathematics (no proving abstract theorems). I saw the Handbook of Linear Algebra as well, but I am concerned it may ...
2
votes
2answers
339 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, ...
1
vote
1answer
26 views

Curl: invariant under change of basis or not?

I wondered how the curl$$\text{rot}\mathbf{F}=\left( \begin{array}{ccc}\partial_y F_3-\partial_z F_2 \\ \partial_z F_1-\partial_x F_3 \\ \partial_x F_2-\partial_y F_1 \end{array} \right)$$of a vector ...
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votes
2answers
24 views

Prove $\exists$ $v \in V$ so that $(v , f(v))$ is a basis of $V$

maybe you guys can help me with this one. Let's say we have a vector space $V$ with $dim(V) = 2$ and we have a linear map $f : V \rightarrow V$ with $f^2 := f \circ f = 0$ ...
0
votes
2answers
31 views

How to determine if matrices are similar?

Trying to teach myself some Linear Algebra, now trying to study about similar matrices concept, but i am having some trouble (maybe because i am trying to teach myself), found a question online and i ...
0
votes
1answer
21 views

Sets of compositions of homomorphisms

I am looking of a relation in the form: $$ Hom(X,Z) = Hom(X,Y)\otimes Hom(Y,Z), $$ or: $$ Hom(X,Z) \subseteq Hom(X,Y)\otimes Hom(Y,Z), $$ or similar (maybe it's not a tensor product? maybe the ...
2
votes
3answers
72 views

Prove that $|GL_n(\mathbb{F})|< q^{n^2}$.

Let $\Bbb F$ be a finite field, say $|\Bbb F|=q$; then we know that $|GL_n(\Bbb F)| < \infty$. But how can we prove that $|GL_n(\mathbb{F})|< q^{n^2}$? I'm guessing because there $n^2$ ...
0
votes
2answers
23 views

Proof equivalence relation

It is given the set $A := \left\{ n \in \mathbb{N} : n \le N, \; N \ne 0 \right\}$. For $a,\,b \in A$ we place $a\text{R}b$ if and only if $a,\,N$ have the same least common multiple of $b,\,N$. Prove ...
0
votes
0answers
41 views

LU factorization

Let $A$ be a nonsingular $n\times n$ matrix and suppose that Gaussian elimination with partial pivoting has been applied to produce $PA = LU$, where : - $P$ is a permutation, - $L$ is a unit lower ...
5
votes
2answers
92 views

Eigenvalues of linear operator $F(A) = AB + BA$

Let $B$ be the $n \times n$ square matrix; $\lambda_1, \lambda_2, \dots, \lambda_n$ are its pairwise distinct eigenvalues. For all $n \times n$ matrix $A$ let me define $F(A) = AB + BA$. We can ...
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votes
0answers
9 views

Linear order question?

I am working on some homework and can someone assist me with these linear order problems? 1.Given a linear order ≤ on Π define s[U,V] ( U,V∈Π ) as the set of all X∈Π such that U≤X≤V or V≤X≤U . Show ...
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0answers
45 views

A condition for surjectivity of a linear map

Let $V,W$ be vector spaces (not necessarily finite dimensional!), and let $W^*$ the dual of $W$. Let $$A:V\longrightarrow W^*$$ be a linear map. What conditions do I have to put on $V$ and especially ...
0
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0answers
31 views

a matrix metric

Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ matrices of the same size. We'll denote $E(U_i,V_i)= \max_v \, |(U_i -V_i)v|$ (max over all the quantum states), $U=\prod_i U_i$ and $V=\prod_i ...
2
votes
0answers
46 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
0
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1answer
101 views

updating of the cholesky decomposition

I try cholrank1 update (wikipedia) of the symmetric positive definite (SPD) matrix . ...
-1
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2answers
17 views

graph quadratic form and find the equation of asymptotes

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes. $$ \left\{ \begin{aligned} ...
5
votes
3answers
69 views

Optimal approximation of quadratic form

Let $\mathbf{x}\in\Bbb{R}^n$ and $A\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes the space of symmetric positive definite $n\times n$ real matrices. Also, let $Q\colon\Bbb{R}^n\to\Bbb{R}_{+}$ be ...
0
votes
1answer
30 views

prove that there exists an upper triangular matrix U such that (U^T)U=A

Let A be a positive definite matrix \begin{pmatrix} a & b \\ b & c \\ \end{pmatrix} prove that there exists an upper triangular matrix U such that U transpose times U equals A. I'm ...
0
votes
1answer
12 views

What is the orthogonal complement of three linearly independent vectors in the 3-dimensional space?

If I have 3 linearly independent vectors, assume the standard basis, in R3, what would be its orthogonal complement? Would there even be one. Isn't the entire space represented by the standard basis?
2
votes
1answer
23 views

Matrix $B \in M_n(S)$, for $S$ an $R$-algebra, with $R$-independent entries, $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent?

Let $R$ be a field (or a domain, or a commutative ring), and $S$ an $R$-algebra. Let $B \in M_n(S)$ have $R$-independent entries. Let $A \in GL_n(R)$. Are the entries of $AB$ $R$-independent? I ...
0
votes
1answer
29 views

For arbitrary subspaces U,V and W of a finite dimensional vectorspace , which of the following relations hold

For arbitrary subspaces U,V and W of a finite dimensional vectorspace , which of the following hold a)U$\cap$(V+ W) $\subset$ U$\cap$V + $U\cap W $ b)U$\cap$(V+ W) $\supset$ U$\cap$V + $U\cap W ...
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2answers
26 views

Orthonormal basis for the null space of almost-Householder matrix

A matrix $H$ is defined as: $$H = I - vv^T$$ where $v$ is a unit vector. What is the rank of $H$? What would be an orthonormal basis for the null space of $H$? How do we find the number of zero ...
12
votes
6answers
6k views

Why does cross product give a vector which is perpendicular to a plane

I was wondering if anyone could give me the intuition behind the cross product of two vectors $\textbf{a}$ and $\textbf{b}$. Why does their cross product $\textbf{n} = \textbf{a} \times \textbf{b}$ ...
0
votes
1answer
33 views

Rapid way to prove $ [e_{ij},e_{lk}]=\delta_{jl}e_{ik}-\delta_{ki}e_{lj} $

Let $e_{ij}$ denote the $n\times n$ matrix with entries all zero but the $(i,j)$th one, in which we put $1$. Let then $\delta_{ij}$ be the Kronecker Delta. Finally $[A,B]:=AB-BA$ is the commutator ...
0
votes
0answers
32 views

Best way to quantify the difference between two vectors

There are plenty of ways of showing an error, or rather a deviation, between two vector quantities. What is the best choice? Specifically, at every timestep, I am comparing two vectors of curvature ...
0
votes
1answer
15 views

Find if a form is symmetric or skew-symmetric

Consider the set of all n × n matrices in R. Given the defined function Φ: $M$(n,n)× $M$(n,n) → R , which Φ(A,B) = $tr$(A$^T$JB) , where J is a skew-symmetric n × n matrix , define if Φ is a ...
0
votes
2answers
37 views

Consider the vector space V = {(a, 1 + a) | a ∈ R} with irregular definitions of addition and multiplication

with addition and scalar multiplication defined by (a, 1 + a) ⊕ (b, 1 + b) = (a + b, 1 + a + b) k '*' (a, 1 + a) = (ka, 1 + ka), k ∈ R find a basis for V. I started off with taking the general ...
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0answers
43 views

How can you do algebra with rounded numbers?

I have a series of seemingly simple algebra problems: 9*x = 5, 5*x = 4, 4*x = 3, 1*x = 1 and ...
1
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3answers
33 views

Eigenvalues of Householder matrix

What would be the eigenvalues for a Householder matrix defined as: $H = I - 2 u u^T$? Can someone explain it to me intuitively or with a simple proof?
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1answer
37 views

Word problem to help me in my math class [on hold]

an estate valued at 124,104 is to be divided between two sons so that the older son receives twice as much as the younger son find each sons share of the estate
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1answer
763 views

Find the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$.

Find a vector equation and parametric equation of the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...