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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Matrix of $f^p:\Lambda^p(E)\rightarrow \Lambda^p(E)$

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

solve problems using linear systems

Leanne works at a greenhouse store. she needs to plant a total of 32 bulbs. two types of bulbs are available. she is asked to plant 3 times as many crocus bulbs as tulip bulbs. how many of each should ...
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0answers
28 views

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row?

How do I generate a sparse invertible 10000 by 10000 binary matrix with 30 to 50 non-zeros per row(uniform distribution)? What sort of algorithm should I use to do this task? Brute Force algorithm- ...
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0answers
23 views

Is the converse of the Spectral Theorem true?

In the book by Friedberg, Insel and Spence, symmetric matrices are orthogonally diagonalizable, and over the complex number field, normal matrices are orthogonally diagonalizable -- this is all from ...
2
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4answers
142 views

If a matrix has positive, real eigenvalues, is it always symmetric?

We know that symmetric matrices are orthogonally diagonalizable and have real eigenvalues. Is the converse true? Does a matrix with real eigenvalues have to be symmetric? A class of symmetric ...
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2answers
28 views

Express $x+y+z$ in terms of $a$ and $b$ [on hold]

If $A = X + Y$ and $B = X + Z$, find the value of $X+Y+Z$ in terms of $A$ and $B$.
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1answer
46 views

Is there a physical interpretation of the alternating property?

A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero. This has statistical significance: regressions on collinear data are bad, dependent ...
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1answer
98 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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5answers
28 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 ...
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1answer
28 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 ...
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0answers
30 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 = ...
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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} + ...
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1answer
31 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 ...
3
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1answer
45 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 ...
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0answers
18 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 ...
0
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0answers
31 views

Solving a 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 $\beta$. ...
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3answers
44 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 ...
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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 $\mathbf{u}(x; y) = (u_1(x;y); u_2(x;y))$, ...
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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 $f^2 := f \circ f = 0$ ...
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2answers
32 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 ...
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1answer
23 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 ...
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2answers
24 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 ...
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0answers
9 views

Linear order question? [on hold]

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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3answers
422 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 ...
2
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1answer
53 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 ...
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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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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
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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 ...
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2answers
95 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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2answers
18 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} ...
0
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1answer
44 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 ...
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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?
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1answer
24 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 ...
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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 ...
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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
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2answers
38 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 ...
3
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1answer
46 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 ...
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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 ...
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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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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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0answers
44 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 ...
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1answer
23 views

Determine the dimension of $U+W$ and of $U \cap W$. Which sums are direct sums?

Problem: Determine the dimension of the sum $U + W$ and of the intersection $U \cap W$ of the following subspaces $U$ and $W$. Which sums are direct sums? 1) $U = \text{span}\left\{(1,1,1)\right\}$ ...
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0answers
35 views

What is the difference between the scalar and vector components of a vector?

What is a scalar component of a vector and what is a vector component of a vector. suppose a vector is making and angle theta with the origin then in my book it is written that its x component is the ...
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0answers
30 views

Functions linearly independent and linearly independent gradients? [on hold]

Let $F_1,...,F_n: \mathbb{R}^n \rightarrow \mathbb{R}$ be a set of $C^{1}$ functions. Is it true that they are linearly independent on a joint level set $\Omega:= \{ p \in \mathbb{R}^n; ...
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0answers
14 views

Relation between bilinear symplectic forms and symplectic matrices

1. Symplectic Forms Let $F : \mathbb{K}^{2n} \times \mathbb{K}^{2n} \to \mathbb{K}$ be a bilinear skew-symmetric nondegenerate form (as known as symplectic form). Then $F(u,v) = u^TAv$ where $A = ...
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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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1answer
29 views

Why the Householder matrix is orthogonal?

A Householder matrix $H = I - c u u^T$, where $c$ is a constant and $u$ is a unit vector, always comes out orthogonal and full rank. Why $H$ is orthogonal (looking for an intuitive proof rather than ...
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0answers
26 views

Question about the coordinates in a new origin on the plane.

I'm reading a book on analytic geometry, specifically on a chapter on change of coordinates. It says that having the origin $O$, one point $P$ and a new origin $O'$, the vector that describes the ...
0
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3answers
30 views

Show that $\ker \hat{T} = \text{ann}(\text{range } T)$

This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by ...
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3answers
49 views

Representing a vector in $\mathbb{R}^{3}$ as sum of only two vectors in $\mathbb{R}^{3}$

Is it possible? Or more generally can any vector in $\mathbb{R}^{n}$ can be represented as sum of (n-1) or less vectors in $\mathbb{R}^{n}$? -----EDIT----- What I basically want to ask is that can ...